Lobachevskii Journal of Mathematics Vol. 1, 1998, 15–44 \begin{matrix}c \\ ◯ \end{matrix}Yu.F. Korobeĭnik

Yu.F. Korobeĭnik

ABSOLUTELY CONVERGENT DIRICHLET SERIES AND ANALYTIC CONTINUATION OF ITS SUM

(submitted by D.Kh.Mustari)

1. Introduction

Two theorems on analytic continuation of the sum of Dirichlet series have been obtained in the paper [1] of T. Kawai, which is the starting point for the present work. These theorems have been proved with the help of convolution operators determined by hyperfunctions supported by the origin. Before formulation of the first result from [1] we introduce the following notations for δ ∈ ℝ; α_{j}, β_{j} ∈ ℝ, j = 1, 2: Ω_{0}(δ) = ℂ, Ω_{0} = Ω_{0}(0) = ℂ; Ω_{j}(δ) = {z ∈ ℂ : α_{j} Im z − β_{j} Re z > δ}, Ω_{l,k}(δ) = Ω_{l}(δ) ⋂ Ω_{k}(δ); Ω_{j} = Ω_{j}(0); Ω_{l,k} = Ω_{l} ⋂ Ω_{k}, k, l = 0, 1, 2; K_{r} = {z : |z| < r}.

Theorem 1 ([1]). Let (λ_{n})_{n=1}^{∞}, (c_{n})_{n=1}^{∞} be sequences of complex numbers. Suppose that the following assumptions are fulfilled:

1) ∀n ≥ 1 λ_{n} ≠ 0, lim_{n→∞} \frac{n}{λ_{n}} = 0;

2) ∃c > 0 ∀n, m ≥ 1 |λ_{n} − λ_{m}| ≥ c|n − m|;

3) there exists a finite number of vectors e_{k}, 1 ≤ k ≤ p, belonging to the unit circumference S^{1} ⊂ ℝ^{2} and such that for each ε > 0 and for each compact K from S^{1} \ (e_{k})_{k=1}^{p} there exists a number n(ε, K) for which

inf\{ |\frac{(Re λ_{n}, Im λ_{n})}{|λ_{n}|} − e| : n ≥ n(ε, K), e ∈ K \} > ε;

4) for some l and k, 0 ≤ l, k ≤ 2, the series

∑_{n=1}^{∞} c_{n} exp(iλ_{n}z)  (1)

converges uniformly on each compact from Ω_{l,k} to the function f(z);

5) the function f(z) has an analytic continuation into the domain Ω_{l,k} ⋃ K_{r} with

r > 0. Then there exists μ > 0 such that f(z) has an analytic continuation to the domain Ω_{l,k}(−μ).

Many-dimensional analogue of Theorem 1 was also obtained in [1].

The method used in this paper differs from the method employed in [1]. It utilizes some well known properties of Dirichlet series described by E. Hille, G. Polya, and the others. A special attention is paid to an effective determination of the domain, into which the sum of the initial series can be continued analytically. In particular, the magnitude of the number μ in Kawai theorem is expressed through the number r. Some methods of effective continuation of the sum of Dirichlet series are also considered including, in particular, the analytic continuation by means of the initial series. In this part of the paper we exploit the results of Leont’ev and other russian mathematicians [2]-[7]. A many-dimensional analogue of Polya theorem is also obtained as well as some results on analytic continuation of the sum of a many-dimensional Dirichlet series. Finally, the characterization of the exact domain of absolute convergence of many-dimensional Dirichlet series is given under comparatively mild restrictions on the exponents of the series.

2. The investigation of one-dimensional series (1)

p.1. According to E. Hille ([8], see also [2, p.194–195]) the set T of all points of absolute convergence of the series (1) is convex and this series converges uniformly on each compact of T. Further, if lim_{n→∞} \frac{ln n}{λ_{n}} = 0, then the maximal domain of convergence of the series (1) coincides with the maximal domain of its absolute convergence (see, for example, [2, p.195–196]). The following important result will be also used frequently in this paper.

Theorem of G. Polya ([9]). Let the series (1) converge in some neighborhood of the point z_{0} ∈ ℂ. Suppose that the condition 1) of theorem 1 holds. Then the sum f(z) of the series (1) is a single-valued function with the convex Weierstrass domain W_{f} of existence.

For an arbitrary domain G ⊂ C^{p}, p > 1 let us denote by H(G) the Fréchet space of all functions analytic in G with the usual topology of uniform convergence on each compact of G.

Let us apply Polya theorem to the following situation. Suppose that the series (1) converges in some domain G and the condition 1) is fulfilled. W.l.o.g. one may assume that the domain G is convex. Let the sum f(z) of the series (1) admit an analytic continuation to the circle K_{r} := { z : |z − z_{0}| < r } where z_{0} is a boundary point of G. According to Polya theorem f ∈ H(conv(G ⋃ K_{r}(z_{0}))). The reader can construct easily the domain conv(G ⋃ K_{r}(z_{0})) for a given convex domain G and z_{0} ∈ ∂G. We consider here only three particular simple types of domain G. Two of them are closely connected with Kawai theorem.

1. Let the series (1) converge in H(Π), where Π is an arbitrary fixed open halfplane. Suppose that the sum f(z) of this series admits an analytic continuation from Π into some circle K_{r}(z_{0}) where z_{0} lies on the boundary line of Π. Using the change of variables z = z_{0} + w exp(iψ) one can replace the series (1) by the similar series

∑_{n=1}^{∞} d_{n} exp iμ_{n}w  (2)

with d_{n} = c_{n} exp iλ_{n}z_{0}, μ_{n} = λ_{n} exp iψ. One can choose ψ in such a manner that the series (2) converges in H(Π_{0}) where ∀r ∈ ℝ Π_{r} := {z : Im z > r}.

So we can always assume that the initial series (1) converges in H(Π_{0}) and its sum f(z) admits an analytic continuation to the circle K_{r} := K_{r}(0). Then conv(Π_{0} ⋃ K_{r}) = Π_{−r}, and by Polya theorem f ∈ H(Π_{−r}).

2. Let the series (1) converge in some strip and let its sum f(z) admit an analytic continuation to the circle K_{r}(z_{0}), z_{0} being a boundary point of this strip. With the help of the same change of variable one can always reduce the situation to the case when the series (1) converges in H(Π_{0,β}) where Π_{α,β} = { z : α < Im z < β } and the sum f(z) of this series admits an analytic continuation to the K_{r}.

If r ≤ β − α, then conv(Π_{0,β} ⋃ K_{r}) = Π_{−r,β}. If r > β − α, then conv(Π_{0,β} ⋃ K_{r}) = Π_{−r,r}. According to Polya theorem f ∈ H(Π_{−r,β}) in the first case and f ∈ H(Π_{−r,r}) in the second one.

3. Suppose that the series (1) converges in H(Δ) where Δ is some angle restricted by two rays l_{1}, l_{2} with the vertex at the origin. Let the sum f(z) admit an analytic continuation to the circle K_{r}. Let T_{j} be the tangent to K_{r} parallel to the ray l_{j} and let \{z_{j}\} := T_{j} ⋂ \overline{K_{r}}. The boundary Γ of the domain conv(Δ ⋂ K_{r}) contains rays Q_{j} lying on the lines T_{j} (j = 1, 2). Each ray Q_{j} begins at the point z_{j} and has the same direction as the ray l_{j}. Besides, Γ contains the boundary arc σ_{r} of the circle K_{r} with the ends at points z_{1}, z_{2} which does not intersect Δ. Simple geometrical arguments show that the domain conv(Δ ⋂ K_{r}) contains the angle Δ_{1} restricted by rays l̃_{j} (j = 1, 2) parallel to l_{j} and such that the distance between the lines on which lie l_{j} and l̃_{j} is equal to r sin γ where 2γ is the magnitude of the angle Γ at the origin.

Let us return now to the first example and suppose that r is the maximal number such that f ∈ H(K_{r}) ( such a number always exists). Then the boundary circumference L of K_{r} contains at least one singular point of f. On the other hand, f ∈ H(Π_{−r}) and all points of L except of −ri belong to Π_{−r}. Consequently z = −ri must be a singular point of f. Under the same supposition on r at least one of two points ±ri in the second example must be a singular point of f. As for the angle Δ in the third example one can only assert that at least one point of the arc σ_{r} is a singular point of f. Examples 1 and 3 give us the following result connected with theorem 1:

Proposition 1. Let suppositions 1), 4), 5) of Kawai theorem 1 be fulfilled. Then the sum f(z) of the series (1) can be extended analytically to the Ω_{l,k}(−μ_{0}) with μ_{0} = r for l = k and μ_{0} = r sin γ if l ≠ k and 2γ is the magnitude of that angle between Ω_{l} and Ω_{k} which is inner one with respect to Ω_{l,k}. Besides, if l = k and r is the maximal number for which f ∈ H(K_{r}), then −ri is the singular point of f.

Proposition 1 shows that the conditions 2) and 3) of the first theorem of T. Kawai are unnecessary. p.2. Polya theorem gives no effective method of analytic continuation of the function f into the domain W_{f}. Such an method can be obtained with the help of one result of A.Leont’ev [3]. We put

L(λ) = ∏_{k=1}^{∞} [1 + (\frac{λ}{λ_{k}} )^{2}] , L_{n}(λ) = ∏_{k=n}^{∞} [1 + (\frac{λ}{λ_{k}} )^{2}] , n = 1, 2, …

It follows from Borel and Lindelőf theorems in the theory of entire functions (see [10, Ch. 1]) that under the condition 1) the functions L(λ), L_{n}(λ) belong to the class of [1, 0] of all entire functions which have minimal type with respect to order 1. The following result of A. Leont’ev is a special case of his more general theorem 3.1.5 in [2, p. 202].

Theorem of Leont’ev . Under assumptions of the theorem of Polya the equality

f(z) = lim_{n→∞} ∑_{k=1}^{n−1} c_{k}L_{n}(iλ_{k}) exp(iλ_{k}z)  (3)

is valid for all z from W_{f}, convergence in (3) being uniform on each compact of W_{f}.

According to this theorem an effective analytic continuation of the function f into the domain conv(G ⋃ K_{r}(z_{0})) can be realized under supposition 1) with the help of relation (3). In particular this relation gives us an effective continuation of the sum f(z) of the series (1) to the domain Ω_{l,k}(−μ_{0}) under suppositions of Proposition 1.

So we obtained the following result.

Proposition 2. If suppositions 1), 4), 5) of theorem 1 take place then the effective analytic extension of the sum f(z) of the series (1) to the domain Ω_{l,k}(−μ_{0}) can be realized by the relation (3).

Again in this situation the conditions 2), 3) are superfluous. p.3. It is also interesting to point out the additional suppositions in Polya theorem under which an analytic continuation of f to W_{f} can be realized by the initial series (1). To do this we introduce firstly one important definition.

The sequence (iλ_{k})_{k=1}^{∞} is said to be effective if each series (1) converging in some H(K_{r}(z_{0})) converges uniformly on every compact of the complete domain of holomorphy W_{f} of its sum f(z). It is evident that if the sequence (iλ_{k})_{k=1}^{∞} is effective and the series (1) converges in H(G) to the function f admitting an analytic continuation to a domain G_{1} containing G then the series (1) converges in H(G_{1}).

Let us consider some examples of effective sequences.

1. Suppose λ_{n} ∈ ℝ, λ_{n} > 0, ∀n ≥ 1, and

lim_{n→∞} \frac{ln n}{λ_{n}} = 0.  (4)

According to old works [8], [11], [12] (see also [2], p. 197, theorem 3.1.3) the series (1) converges absolutely and uniformly on each compact of the halfplane Π_{γ} where γ = lim sup_{k→∞} \frac{ln |c_{k}|}{λ_{k}}, and diverges in the halfplane Im z < γ. Suppose that numbers

{λ_{n}}_{n=1}^{∞} satisfy the condition 1) of Kawai theorem which is more restrictive than (4). We put

L(λ) = ∏_{m=1}^{∞} (1 + (\frac{λ}{λ_{m}} )^{2}) , δ_{1} = lim sup_{m→∞} \frac{1}{|λ_{m}|} ln \frac{1}{|L′(λ_{m})|}.

As it was mentioned above L(λ) ∈ [1, 0].

Proposition 3. If δ_{1} = 0 and condition 1) holds, then the sequence {iλ_{k}}_{k=1}^{∞} is effective.

Indeed, according to theorem of V. Bernstein ( [11]; see also [2, p. 174]) under suppositions of Proposition 3 each point on the boundary line Im z = γ of the halfplane Π_{γ} of absolute convergence of the series (1) is a singular point of its sum f. Hence W_{f} = Π_{γ}.

2. Suppose now that numbers λ_{n} satisfy the following conditions

a) lim_{n→∞} arg λ_{n} = 0; b) lim sup_{n→∞} \frac{n}{|λ_{n}|} < ∞.

Then according to the lemma on page 150 of [3] the series (1) with lim sup_{n→∞} \frac{ln |c_{k}|}{|λ_{n}|} =

b ∈ (−∞, +∞), converges in H(Π_{l}) and diverges everywhere in Im z < b. Further ([3], p. 83) if conditions a) and 1) hold and, besides, if

lim inf_{n→∞}(|λ_{n+1}| − |λ_{n}|) > 0,  (5)

then each point on the line Im z = b is a singular point of f. So W_{f} = Π_{b}, and the sequence {iλ_{k}}_{k=1}^{∞} is effective. Polya theorem leads to the following result in this situation.

Proposition 4. Let {λ_{n}}_{n=1}^{∞} be complex number satisfying the conditions 1), a) and (5). Suppose that the series (1) converges in H(Π_{0}) and its sum f can be extended analytically to the circle K_{r}. Then the series (1) converges in H(Π_{−r}). Besides, if r is the maximal radius for which f ∈ H(K_{r}), then each point of the line Im z = −ir is a singular point of f.

It is worth mentioning that if λ_{n} ∈ ℝ, λ_{n} > 0 and condition (5) holds, then [11] δ_{1} = 0. p.4. Let us give some additional remarks for the case λ_{k} ∈ ℝ, k ≥ 1 which will be used later in many-dimensional situation. Suppose

lim_{n→∞} \frac{ln n}{λ_{n}} = 0.  (6)

Then, according to above mentioned papers [8], [11], [12] the series (1) converges absolutely and uniformly on each compact of some strip Π_{γ,β} = { z : −∞ ≤ γ < Im z < β ≤ +∞ }, and diverges everywhere outside the closure ClΠ_{γ,β} of Π_{γ,β} in ℂ. We put N_{1} = { n ≥ 1 : λ_{n} ≥ 0 }, N_{2} = { n ≥ 1 : λ_{n} < 0 }. It is easy to check that the series ∑_{n∈N_{1}} c_{n} exp(iλ_{n}z) converges absolutely and uniformly on each compact of

the halfplane Π_{γ}, while the second series ∑_{n∈N_{2}} c_{n} exp(iλ_{n}z) behaves itself in the same

way in the halfplane Im z < β. W.l.o.g. one can assume γ = 0. So suppose that the series (1) converges in the strip Π_{0,β}, and its sum is equal to f(z). For all z ∈ Π_{0,β} we have f(z) = f_{1}(z) + f_{2}(z); f_{j}(z) = ∑_{n∈N_{j}} c_{n} exp(iλ_{n}z), j = 1, 2.

It is clear that investigation of the convergence of the series (1) and of analytic continuation of its sum near the line Im z = 0 needs only consideration of the series ∑_{n∈N_{1}} c_{n} exp(iλ_{n}z) and of its sum f_{1}(z).

The following results are direct corollaries of the previous propositions.

Proposition 5. Suppose λ_{m} ∈ ℝ for all m ≥ 1 and condition (6) is fulfilled. Let the sequences { iλ_{n_{m}} : m = 1, 2, …; n_{m} ∈ N_{1} } and { iλ_{k_{m}} : m = 1, 2, …; k_{m} ∈ N_{2} } be effective. Suppose that the series (1) converges in the strip Π_{0,β}, 0 < β ≤ ∞. Then the series (1) converges absolutely and uniformly on each compact of the maximal strip Π_{α,δ} of holomorphy of its sum f(z), where δ ≥ β, α ≤ 0.

Proposition 6. Suppose λ_{m} ∈ ℝ, m ≥ 1 and condition (6) holds. Let the sequences { iμ_{m} }_{m=1}^{∞} : μ_{m} = λ_{n_{m}}, n_{m} ∈ N_{1}, ∀m ≥ 1, be effective. Suppose that the series (1) converges in the strip Π_{0,β}, 0 < β ≤ ∞, and its sum f(z) admits an analytic continuation into the circle |z| < r. Then the series (1) converges absolutely and uniformly on each compact of the strip Π_{−r,β}.

Proposition 7. Suppose that ∀n ≥ 1 λ_{n} ∈ ℝ, and conditions 1) and (5) are fulfilled. Let the series (1) converge in the strip Π_{0,β} and let its sum f(z) admit an analytic continuation to the circle |z| < r. Then the series (1) converges absolutely and uniformly on each compact of the strip Π_{−r,β}

3. Strictly effective sequences

p.1. Let z_{0} ∈ ℂ and let H{z_{0}} be the space of all analytic germs at the point z_{0} with the usual inductive topology [13]. It is well known that H{z_{0}} is a LN^{∗}-space in the sense of Sebastiao e Silva ([14], ch. XI), and that the series (1) converges in H{z_{0}} iff it converges in H(K_{ρ}) with some ρ > 0.

For an arbitrary sequence { μ_{k} }_{k=1}^{∞} of pairwise different complex numbers denote by (exp μ_{k}z; z_{0}) the closure of the linear hull of the set { exp μ_{k}z : k = 1, 2, … } in the space H{z_{0}}. According to [14] (exp μ_{k}z; z_{0}) is also a LN^{∗}-space.

The sequence { iλ_{k} }_{k=1}^{∞} is said to be strictly effective if for each z_{0} ∈ ℂ every function v from (exp iλ_{l}z; z_{0}) can be represented by the series (1) converging in H(W_{v}), and such a representation is unique.

It follows from this definition that each strictly effective sequence is effective.

As above denote by [1, 0] the class of all entire functions of minimal type with respect to order 1 (the functions of infraexponential type). The following result is the main one in this paragraph.

Proposition 8. Let Δ := (iλ_{k})_{k=1}^{∞} be an arbitrary sequence of pairwise different complex numbers satisfying the condition 1) of theorem 1. The sequence Δ is strictly effective iff the interpolation problem (in. pr.)

y(iλ_{k}) = d_{k},  k = 1, 2, …  (7)

has a solution in [1, 0] for each sequence (d_{k})_{k=1}^{∞} such that

lim_{k→∞} sup \frac{1}{|λ_{k}|} ln |d_{k}| ≤ 0.  (8)

p.2. Before beginning the proof of proposition 8 let us investigate some properties of the space (exp iλ_{l}z; z_{0}). It is evident that this space is invariant with respect to differentiation. It is also well known that the spectrum E_{Λ}(z_{0}) of (exp iλ_{l}z; z_{0}) is the set of all pairs (μ, k), μ ∈ ℂ, k ≥ 1, such that z^{m} exp μz ∈ (exp iλ_{l}z; z_{0}), ∀m ≤ k − 1, but z^{k} exp μz ∉ (exp iλ_{l}z; z_{0}). Let us show that E_{Λ}(z_{0}) coincides with the set { (iλ_{k}, 1) : k = 1, 2, … }.

It is evident that exp iλ_{k}z ∈ (exp iλ_{l}z; z_{0}), ∀k ≥ 1. If L(x) = ∑_{k=0}^{∞} a_{k}λ^{k} is an arbitrary function from the class [1, 0] and L(D)y := ∑_{k=0}^{∞} a_{k}y^{(k)}(z), then [15] the operator L(D) is continuous in H{z_{0}} for all z_{o} ∈ ℂ and in H(G) for an arbitrary domain G in ℂ.

Suppose that z exp iλ_{k_{0}}z ∈ (exp iλ_{l}z; z_{0}) for some k_{0} ≥ 1. Then uniformly in some circle \overline{K_{r}(z_{0})} := \{ z : |z − z_{0}| ≤ r \} with r > 0:

z exp iλ_{k_{0}}z = lim_{n→∞} ∑_{k=1}^{m_{n}} b_{k,n} exp iλ_{l}z.

Applying to this equality the operator L(D) we obtain for all z ∈ K_{r}(z_{0})

L(D)(z exp iλ_{k_{0}}z) = lim_{n→∞} ∑_{k=1}^{m_{n}} b_{k,n}L(D)(exp iλ_{l}z).  (9)

The left-hand part of the equality (9) is equal to

∑_{k=0}^{∞} a_{k}(z exp iλ_{k_{0}}z)^{(k)} = L(iλ_{k_{0}})z exp iλ_{k_{0}}z + L′(iλ_{k_{0}}) exp iλ_{k_{0}}z.

Let us choose the function L(λ) in such a manner that all iλ_{k} be simple roots of L. Namely, if λ_{1} ≠ 0 (we suppose that |λ_{k}| ≤ |λ_{k+1}|, ∀k ≤ 1), we can select

L, following [2], in the form L_{0}(λ) = ∏_{k=1}^{∞}′(1 + \frac{λ^{2}}{λ_{k}^{2}} ), where the product is taken over

those ∀k ≥ 1 for which all numbers λ_{k}^{2} are pairwise different. If λ_{1} = 0, we take

L_{0}(λ) = λ ∏_{k=2}^{∞}′(1 + \frac{λ^{2}}{λ_{k}^{2}} ). In both cases L_{0}(λ) ∈ [1, 0] and iλ_{k} are simple roots of

L_{0}. W.l.o.g. we can assume then λ_{k} ≠ 0, ∀k ≥ 1. The function L_{0}(λ) having been constructed, we obtain that the left-hand part of the equality (9) is equal to L′_{0}(iλ_{k_{0}}) exp iλ_{k_{0}}z ≠ 0 for all z ∈ ℂ, while the right-hand part of (9) is equal identically to zero. This contradiction shows that z exp iλ_{k}z ∉ (exp iλ_{l}z; z_{0}), ∀k ≥ 1. Therefore (iλ_{k}, 1) ∈ E_{Λ}(z_{0}), ∀k ≥ 1. Suppose now that ∃μ ∈ ℂ : μ ≠ ±iλ_{k}, k ≥ 1, exp μz ∈ (exp iλ_{l}z; z_{0}). Then uniformly in some circle |z − z_{0}| < d

exp μz = lim_{n→∞} ∑_{k=1}^{m_{n}} d_{k,n} exp λ_{k}z.

Hence ∀z ∈ K_{d}(z_{0}).

L_{0}(D)(exp μz) = lim_{n→∞} ∑_{k=1}^{m_{n}} d_{k,n}L_{0}(D)(exp iλ_{k}z) ≡ 0.

But L_{0}(D)(exp μz) = L_{0}(μ) exp μz ≠ 0. Hence exp μz ∉ (exp iλ_{l}z; z_{0}), and (μ, k) ∉ E_{Λ}(z_{0}), ∀k ≥ 1.

Suppose at last that ∃k_{1} ≥ 1 : exp(−iλ_{k_{1}}z) ∈ (exp iλ_{l}z; z_{0}). We put L_{(1)}(λ) = \frac{L_{0}(λ)}{(λ^{2}+λ_{k_{1}}^{2})}. Then L_{(1)}(λ) ∈ [1, 0] by category theorem ([10], ch. I, §9). We have

uniformly in some circle |z − z_{0}| ≤ ρ

exp(−iλ_{k_{1}}z) = lim_{n→∞} ∑_{k=1}^{m_{n}} b_{k,n} exp iλ_{k}z.

Hence L_{(1)}(D)[exp(−iλ_{k_{1}}z)] = lim_{n→∞} ∑_{k=1}^{m_{n}} b_{k,n}L_{(1)}(D)(exp iλ_{k}z) and

L_{(1)}(−iλ_{k_{1}}) exp(−iλ_{k_{1}}z) = lim_{n→∞} b_{k,n}L_{(1)}(iλ_{k_{1}})(exp iλ_{k_{1}}z).

Since L_{(1)}(−iλ_{k_{1}}) = L_{(1)}(iλ_{k_{1}}) ≠ 0, we have exp(−2iλ_{k_{1}}z) = lim_{n→∞} b_{k_{0},n} = B_{k_{0}}, and

we again obtain a contradiction.

So the spectrum of (exp iλ_{l}z; z_{0}) coincides with the set {(λ_{k}, 1) : k ≥ 1} (such a spectrum is called simple). We say that the set Q of the space H admits spectral synthesis in H if Q coincides with the closure of linear hull of all Dirichlet monomials z^{m} exp λz, containing in Q. If is evident that the set (exp iλ_{l}z; z_{0}) admits spectral synthesis in H{z_{0}}. Besides, since exp iλ_{k}z ∈ (exp iλ_{l}z; z_{0}) for all k ≥ 1, but exp μz ∉ (exp iλ_{l}z; z_{0}), if μ ≠ iλ_{k}, k = 1, 2, …, the space (exp iλ_{l}z; z_{0}) is a nontrivial proper subspace of H{z_{0}} with simple spectrum, invariant with respect to differentiation and admitting spectral synthesis in H{z_{0}}.

We shall also use in the proof of Proposition 8 some results from [2]. Following [2], we put ∀k ≥ 1 L_{k}(λ) = \frac{L_{0}(λ)}{(λ−iλ_{k})L′_{0}(iλ_{k})}. Then L_{k} ∈ [1, 0], ∀k ≥ 1. Let ψ_{k}(t) be the function associated with L_{k} in the sense of Borel:

L_{k}(λ) = ∑_{n=0}^{∞} a_{k,n}λ^{n}; ψ_{k}(t) = ∑_{n=0}^{∞} a_{k,n}n!t^{−n−1}, k = 1, 2, ….

The functions ψ_{k}(t) are analytic in the domain |t| > 0 and vanish at infinity. We have for each closed rectifiable Jordan curve Γ such that 0 ∈ int Γ

L_{k}(λ) = \frac{1}{2πi} ∫_{Γ} ψ_{k}(t) exp λt dt, k = 1, 2, …

It is evident, that for all k, m ≥ 1

\frac{1}{2πi} ∫_{Γ} ψ_{m}(t) exp iλ_{k}t dt = L_{m}(iλ_{k}) = δ_{k,m},

where δ_{k,m} is the Kronecker symbol. Hence, if G is an arbitrary simply connected domain in ℂ containing the origin, then the system { ψ_{m}(z) }_{m=1}^{∞} is biorthogonal to the system { exp iλ_{k}z }_{k=1}^{∞} [2]. This fact implies that if v(z) is the sum of the series

v(z) = ∑_{m=1}^{∞} d_{m} exp(iλ_{m}z)  (10)

converging in H(G), then such a representation of the function v is unique. Note that inclusion 0 ∈ G can be achieved by the linear change of variable: z = w + z_{0} which does not alter the form of the series (10). Therefore the uniqueness of the representation (10) converging in H(G) is valid for an arbitrary simply connected domain G in ℂ.

One simple remark will be also useful in the proof. Suppose that each function v(z) from (exp λ_{k}z; 0) can be represented in the form of the series (10) converging in H(W_{v}). Let z_{0} ∈ ℂ and g ∈ (exp λ_{k}z; z_{0}). We put h(t) = g(t + z_{0}). It is evident that h ∈ H({0}) and z_{0} + W_{h} = W_{g}. Since g ∈ (exp λ_{l}z; z_{0}), we have uniformly

in |z − z_{0}| ≤ r with r > 0 g(z) = lim_{n→∞} ∑_{k=1}^{m_{n}} b_{k,n}e^{iλ_{k}z}, or h(t) = lim_{n→∞} ∑_{k=1}^{m_{n}} d_{k,n}e^{iλ_{k}t},

d_{k,n} = b_{k,n}e^{iλ_{k}z_{0}}, convergence being uniform in |t| ≤ r. Hence h ∈ (exp λ_{k}z; 0) and therefore

h(t) = ∑_{k=1}^{∞} f_{k} exp iλ_{k}t,  (11)

the series converging in H(W_{h}). If we put in (11) t = z − z_{0}, then

g(z) = ∑_{k=1}^{∞} f_{k} exp(−iλ_{k}z_{0}) exp iλ_{k}z,

and this series converges in H(W_{g}). Besides, if the representation of each function v from (exp λ_{k}; 0) in the form of the series (10) converging in H(W_{v}) is unique, the same assertion is valid for representation of each function g from (exp λ_{k}z; z_{0}) in the form of the series (10) converging in H(W_{g}). Consequently, the sequence { iλ_{k} }_{k=1}^{∞} is strictly effective iff each function v from H({0}) can be represented uniquely in the form of the series (1) converging in H(W_{v}). p.3. After all these preliminary arguments we can pass on to the proof of proposition 8 which is based on two following theorems from [4]. For the reader’s convenience we give here their formulations in one special case which will be used in the proof. Namely, if a closed bounded convex set D in Proposition 7 and in theorem 23 in [4] coincides with the origin and numbers p_{n} in theorem 23 are equal to zero, then we have the following results.

Theorem K_{1} (Proposition 7, [4]). Let W be a nontrivial closed invariant with respect to differentiation subspace of H{0} with simple spectrum (iλ_{k})_{k=1}^{∞}. Suppose that W admits spectral synthesis in H{0}. Then the following assertions are equivalent:

1) (exp iλ_{k}z)_{k=1}^{∞} is a basis in (exp λ_{k}; 0).

2) Any function f in W can be represented in a unique way in the form of series (1) converging in H(W_{f}).

Theorem K_{2} (Theorem 23, [4] ). The system ( exp iλ_{k}z)_{k=1}^{∞} constitutes a basis in ( exp iλ_{k}z; 0 ) iff the in. pr. (7) has a solution in [1, 0] for each sequence {d_{k}}_{k=1}^{∞} satisfying the condition (8).

If we put in theorem K_{1} W = (exp iλ_{k}z; 0), then we obtain that any function f from (exp λ_{l}z; 0) can be represented in a unique way by the series (1) converging in H(W_{f}) iff the in. pr. (7) is solvable in the class [1, 0] for each number sequence { d_{k} }_{k=1}^{∞} satisfying the condition (8). This completes the proof of Proposition 8.

Corollary . If the in. pr. (7) has a solution in [1, 0] for any number sequence {d_{k}}_{k=1}^{∞} satisfying (8), then the sequence {iλ_{k}}_{k=1}^{∞} is effective.

According to [5] the interpolation problem (7) has a solution in [1, 0] for each sequence (d_{k})_{k=1}^{∞} satisfying (8) iff

1) lim_{k→∞} \frac{k}{λ_{k}} = 0;

2) ∀δ ∈ (0, 1) lim_{k→∞} sup \frac{1}{|λ_{k}|} ln Φ_{k}^{δ} ≤ 0,

Φ_{k}^{δ} = ∏_{{j≠k \atop |λ_{j} −λ_{k}|≤δ|λ_{k}|}} |1 − \frac{λ_{j}}{λ_{k}} |.

It follows from the results of [6], [7] that the condition 2) is equivalent to

2′)  lim_{δ→0} lim sup_{k→∞} \frac{1}{|λ_{k}|} ∫_{0}^{δ|λ_{k}|}\frac{n(iλ_{k}, t) − 1}{t} dt = 0,

where n(μ, t) is the number of points {λ_{j}}_{j=1}^{∞} belonging to the circle { z : |μ−z| ≤ t }.

We give now a simple condition providing 2′). Let ϕ(t) be a function defined on [0, +∞) such that

ϕ(t) = 0, 0 ≤ t < 1, ϕ(1) = γ > 0,

and ϕ(t) is a continuous strictly increasing function on [1, +∞). We put ψ(t) := ϕ^{−1}(t) and suppose that

lim_{t→∞} sup \frac{t}{ϕ(t)} < ∞,

that is there exists B < ∞ such that ∀t ∈ [γ, +∞) ψ(t/B) ≤ t. Suppose also that

inf\{ \frac{|λ_{n} − λ_{s}|}{ϕ(|n − s|)} : n, s ≥ 1, n ≠ s \} = d > 0  (12)

If s ≠ m and |iλ_{m} − iλ_{s}| ≤ t, then dϕ(|m − s|) ≤ t. Hence for all t ≥ dγ we have n(iλ_{m}, t) ≤ 2ψ (\frac{t}{d}), and if m ≥ N_{0}, then

∫_{0}^{δ|λ_{m}|}\frac{n(iλ_{m}, t) − 1}{t}dt = ∫_{dγ}^{δ|λ_{m}|}\frac{n(iλ_{m}, t) − 1}{t}dt

< ∫_{dγ}^{δ|λ_{m}|}\frac{n(iλ_{m}, t)}{t}dt ≤ 2 ∫_{dγ}^{δ|λ_{m}|}\frac{ψ (\frac{t}{d})}{t}dt ≤ \frac{2Bδ|λ_{m}|}{d},

which implies 2′).

For example, we can take ϕ(t) = 0, 0 ≤ t < 1, and ϕ(t) = ct or ϕ(t) = ct ln t, c > 0, when t ∈ [1, +∞).

We shall say that condition K is fulfilled for the sequence { iλ_{n} }_{n=1}^{∞} if the relation (12) holds where ϕ(t) = 0, 0 ≤ t < 1, ϕ(1) > 0, ϕ(t) is a continuous strictly increasing function on [1, +∞) such that lim_{t→∞} sup \frac{t}{ϕ(t)} < ∞.

We can formulate now the following addition to Polya and Kawai theorems.

Proposition 9. Let the sequence { iλ_{n} }_{n=1}^{∞} satisfy the conditions 1) and 2′). Then the series (1) converges absolutely and uniformly on each compact of the domain W_{f} under the suppositions of Polya theorem and of the domain Ω_{l,k}(−μ_{0}) under suppositions 4), 5) of Kawai theorem.

Corollary . If the sequence { iλ_{k} }_{k=1}^{∞} satisfies the conditions 1) and K, then all assertions of Proposition 9 are valid.

It is evident that Proposition 9 and its corollary give us results more strong than theorem 1 of Kawai. p.4. A criterion for the sequence { iλ_{k} }_{k=1}^{∞} being effective is given by proposition 2 in [16]. The proof of that proposition is similar to the proof of the proposition 8 in this paper and is based on theorems K_{1} and K_{2} applied to the space T_{R} of all series (1) converging in H(K_{R}) (0 < R < ∞). It is proved in [16] that for each R ∈ (0, +∞) the space T_{R} has a simple spectrum { (iλ_{k}, 1) : k = 1, 2, … }. It is evident that T_{R} is invariant with respect to differentiation. But the closeness of T_{R} was not proved in [16] which made the proof incorrect. It is evident that span(e^{iλ_{k}z})_{k=1}^{∞} ⊆ T_{R} ⊆ (e^{iλ_{k}z}; H(K_{R})), 0 < R < ∞. Hence closure of T_{R} coincides with (e^{iλ_{k}z}; H(K_{R})), and T_{R} is closed for all R ∈ (0, +∞) iff T_{R} = (e^{iλ_{k}z}; H(K_{R})), 0 < R < ∞. This means that if we eliminate the incorrectness in the proof of proposition 2 in [16] by adding some assumption providing the closeness of T_{R} that is the equality T_{R} = (e^{iλ_{k}z}; H(K_{R})), then we shall obtain finally the result which will be contained in proposition 8 of this paper.

It is worth noting that though the validity of proposition 2 in [16] is not proved and moreover it seems now to be doubtful nevertheless assertions of proposition 3 in [16] are correct due to proposition 8 of this paper.

4. Many-dimensional Dirichlet series

p.1. Passing to many-dimensional Dirichlet series we remind first of all some their properties which are similar to relevant properties of one-dimensional series.

Lemma 1. The set Q of all points of absolute convergence of series

∑_{n=1}^{∞} c_{n} exp(∑_{k=1}^{p} λ_{n,k}z_{k}),  λ_{n,k} ∈ ℂ, z_{k} ∈ ℂ, p ≥ 1,  (13)

is convex. The series (13) converges absolutely in H(int Q).

Proof. Let A(z_{1}, z_{2}, …, z_{p}) ∈ Q, B(w_{1}, w_{2}, …, w_{p}) ∈ Q, α ∈ (0, 1), D_{α} = (αz_{1} + (1 − α)w_{1}, …, αz_{p} + (1 − α)w_{p}).

We have for n ≥ 1

τ_{n} := |c_{n} exp[∑_{j=1}^{p} λ_{n,j}(αz_{j} + (1 − α)w_{j})]|

= |c_{n} exp[∑_{j=1}^{p} λ_{n,j}z_{j}]|^{α} · |c_{n} exp[∑_{j=1}^{p} λ_{n,j}w_{j}]|^{1−α} .

According to the well known inequality (see, for example, [2, p. 194]), if A_{1} ≥ 0, A_{2} ≥ 0, then (A_{1})^{α}(A_{2})^{1−α} ≤ A_{1} + A_{2}, ∀α ∈ [0, 1].

Hence for all n ≥ 1

τ_{n} ≤ |c_{n} exp(∑_{j=1}^{p} λ_{n,j}z_{j})| + |c_{n} exp(∑_{j=1}^{p} λ_{n,j}w_{j})| ,

and D_{α} ∈ Q.

Let us fix an arbitrary point z_{0}(z_{1,0}, …, z_{p,0}) from int Q. One can find q points z^{(k)}(z_{1,k}, …, z_{p,k}), k = 1, 2, …, q, from int Q such that z_{0} ∈ int B_{0} and B_{0} ⊂ int Q, where B_{0} is the convex polyhedron (simplex) restricted with the surface (frontier)

S_{0} = { z(z_{1}, …, z_{p}), z_{j} = ∑_{s=1}^{q} μ_{s}z_{j,s} : 0 ≤ μ_{s} ≤ 1, ∑_{s=1}^{q} μ_{s} = 1 }.

It is easy to prove by induction the inequality

∏_{k=1}^{n} (A_{k})^{γ_{k}} ≤ ∑_{k=1}^{n} A_{k},

where n ≥ 1, 0 ≤ γ_{k} ≤ 1, 0 ≤ A_{k}, k = 1, 2, …, n, ∑_{k=1}^{n} γ_{k} = 1.

For all n ≥ 1 we have

|c_{n}| max { |exp(∑_{j=1}^{p} λ_{n,j}z_{j})| : z ∈ S_{0} } =

= |c_{n}| max{ ∏_{s=1}^{q} |exp(∑_{j=1}^{p} λ_{n,j}z_{j,s})|^{μ_{s}} : 0 ≤ μ_{s} ≤ 1, ∑_{s=1}^{p} μ_{s} = 1 } ≤

≤ ∑_{s=1}^{q}|c_{n} exp(∑_{j=1}^{p} λ_{n,j}z_{j,s})| =: δ_{n}

where

∑_{n=1}^{∞} δ_{n} < +∞.

We can take a sufficiently small δ > 0 such t hat K_{δ}(z_{0}) ⊂ int B_{0}, where K_{δ}(z_{0}) = {z : |z − z_{0}|_{p} < δ} and

|z − z_{0}|_{p} = (∑_{k=1}^{p} |z_{k} − z_{k,0}|^{2})^{\frac{1}{2}} .

By maximum principle

(n ≥ 1) |c_{n}| max { |exp(∑_{j=1}^{p} λ_{k,j}z_{j})| : z ∈ K_{δ}(z_{0}) } ≤ δ_{n}.

The last inequality and Heine-Borel lemma imply the absolute convergence of the series (13) in H(int Q). □

Corollary 1. If G is an arbitrary convex domain in ℂ^{p}, then the following assertions are equivalent:

1) the series (13) converges absolutely at each point of G;

2) the series (13) converges absolutely in H(G).

Corollary 2. If the series (13) converges absolutely in the neighborhood of some point z_{0} of ℂ^{p} then there exists the maximal domain abc(13) of absolute convergence of the series (13). This domain is convex (namely, abc(13) = int Q).

Lemma 2. Let the series (13) satisfy the following condition

lim_{n→∞} \frac{ln n}{∑_{k=1}^{p} |λ_{n,k}|} = 0.  (14)

Then the maximal domain of convergence of the series (13) coincides with the maximal domain of its absolute convergence.

Proof. Let the series (13) converge in some domain G ⊂ ℂ^{p}. Having fixed an arbitrary z_{0} ∈ G we can always select the polyhedron B_{0} with the frontier S_{0} in such a manner that z_{0} = (z_{1,0}, …, z_{p,0}) ∈ int B_{0}, B_{0} ⊂ G,

S_{0} = { z = (z_{1}, …, z_{p}), z_{j} = ∑_{k=1}^{q} μ_{k}z_{j,k} : 0 ≤ μ_{k} ≤ 1, ∑_{k=1}^{q} μ_{k} = 1 }.

Let us take a sufficiently small sphere K_{δ}(z_{0}) such that K_{δ}(z_{0}) ⊂ int B_{0}. As above, the maximum principle implies ∀z ∈ K_{δ}(z_{0}) ∀n ≥ 1

|c_{n} exp(∑_{j=1}^{p} λ_{n,j}z_{j})| ≤ ∑_{s=1}^{q} |c_{n} exp(∑_{j=1}^{p} λ_{n,j}z_{j,s})| ≤ M < +∞.

If z̃_{j,n} = z_{j,0} + \frac{δ}{2\sqrt{p}} exp(−i arg λ_{n,j}), j = 1, 2, …, p, then z̃_{n} = (z̃_{1,n}, …, z̃_{p,n}) ∈ K_{δ}(z_{0}) and

M ≥ |c_{n} exp(∑_{j=1}^{p} λ_{n,j}z̃_{j,n})| =

= |c_{n} exp(∑_{j=1}^{p} λ_{n,j}z_{j,0})| · |exp(∑_{j=1}^{p} λ_{n,j}(z̃_{j,n} − z_{j,0}))| =

= |c_{n} exp(∑_{j=1}^{p} λ_{n,j}z_{j,0})| · exp(\frac{δ}{2\sqrt{p}} ∑_{j=1}^{p} |λ_{n,j}|).

Thus ∀n ≥ 1

|c_{n}||exp ∑_{j=1}^{p} λ_{n,j}z_{j,0}| ≤ M exp(−\frac{δ}{2\sqrt{p}} ∑_{j=1}^{p} |λ_{n,j}|) =: h_{n},

where ∑_{n=1}^{∞} h_{n} < ∞ by condition (14). Consequently the series (13) converges abso-

lutely at the point z_{0}. □

Corollary . Let G be an arbitrary convex domain in ℂ^{p} and let condition (14) be fulfilled. Then each of assertions 1), 2) is equivalent to any of two following assertions:

3) the series (13) converges at each point in G;

4) the series (13) converges in H(G).

p.2. The remaining part of this paragraph is connected with the following problem. Suppose that the series (13) converges uniformly on each compact of some domain D ⊆ ℂ^{p} and its sum f(z_{1}, …, z_{p}) admits an analytic continuation to the domain G containing D. We shall look for additional assumptions under which the initial series (13) converges in some domain G_{1} such that D ⊂ G_{1} ⊆ G.

Let us begin from the most simple case when the domains D and G are Cartesian products of one-dimensional domains.

Proposition 10. Let D_{j} and G_{j} be nonempty domains in ℂ such that D_{j} ⊆ G_{j},

j = 1, …, p. Let the series (13) converge in D := ×_{j=1}^{p} D_{j}. Suppose that the sum f(z_{1}, …, z_{p}) of the series (13) admits an analytic continuation from D to the domain

G := ×_{j=1}^{p} G_{j}.

Finally, let the following conditions be fulfilled:

A) all sequences { λ_{n,j} }_{n=1}^{∞}, j = 1, …, p, are effective, and the equalities

lim_{n→∞} \frac{ln n}{λ_{n,j}} = 0, j = 1, 2, …, p  (15)

hold.

Then the initial series (13) converges absolutely in H(B), where B := ×_{j=1}^{p} G_{j}^{c} and

G_{j}^{c} := conv G_{j}, j = 1, 2, …, p.

Proof. Due to lemmas 1–2 the series (13) converges absolutely in H(D). Hence f(z_{1}, …, z_{p}) ∈ H(D). It is evident that proposition 10 is valid for the case p = 1. Assuming that the proposition is true for all p ≤ p_{0}, we shall prove its validity for p = p_{0} + 1 (p_{0} ≥ 1). Let us put in (13) p = p_{0} + 1 and fix an arbitrary z_{p_{0}+1} from D_{p_{0}+1}. If

a_{n} = c_{n} exp λ_{n,p_{0}+1}z_{p_{0}+1}, n = 1, 2, …

then the series

∑_{n=1}^{∞} a_{n} exp ∑_{j=1}^{p_{0}} λ_{n,j}z_{j}  (16)

converges absolutely in H(×_{j=1}^{p_{0}} D_{j}) and its sum

h(z_{1}, …, z_{p}) := f(z_{1}, z_{2}, …, z_{p_{0}}, z_{p_{0}+1})

admits an analytic continuation into the domain ×_{j=1}^{p_{0}} G_{j}. By inductive hypothesis

the series (16) converges absolutely in H(A), where

A = ×_{j=1}^{p_{0}} G_{j}^{c}.

Hence the initial series (13) converges absolutely in the domain A × D_{p_{0}+1}. Now let

us fix an arbitrary points z_{1}, …, z_{p_{0}} in the domain ×_{j=1}^{p_{0}} G_{j} and put

b_{n} = c_{n} exp ∑_{j=1}^{p_{0}} λ_{n,j}z_{j}, n ≥ 1.

The series

∑_{n=1}^{∞} b_{n} exp λ_{n,p_{0}+1}z_{p_{0}+1}  (17)

converges absolutely in D_{p_{0}+1}, and its sum

g(z_{p_{0}+1}) := f(z_{1}, …, z_{p_{0}}, z_{p_{0}+1})

admits an analytic continuation into the domain G_{p_{0}+1}. Hence the series (17) converges absolutely in H(G_{p_{0}+1}^{c}). The initial series converges absolutely in the domain

×_{j=1}^{p_{0}} G_{j} × G_{p_{0}+1}^{c}.

If f(z_{1}, …, z_{p_{0}}, z_{p_{0}+1}) is the sum of the series (13) absolutely converging in the domain

Q = (×_{j=1}^{p_{0}} G_{j}) × G_{p_{0}+1}^{c},

then by lemmas 1–2 the series (13) converges absolutely in H(Q), hence

f(z_{1}, …, z_{p_{0}}, z_{p_{0}+1}) ∈ H(Q).

Let us fix an arbitrary z̃_{p_{0}+1} from G_{p_{0}+1}^{c} and put

b(z_{1}, …, z_{p_{0}}) = f(z_{1}, …, z_{p_{0}}, z̃_{p_{0}+1}).

The series (16) with a_{n} = c_{n} exp λ_{n,p_{0}+1}z̃_{p_{0}+1}, n ≥ 1, converges absolutely in the

domain ×_{j=1}^{p_{0}} G_{j}, and its sum b(z_{1}, …, z_{p_{0}}) is analytic in this domain. Reutilizing the

inductive hypothesis we obtain that the series (16) converges absolutely in H(A). Then the initial series (13) converges absolutely in the domain A × G_{p_{0}+1}^{c}, and according to lemma 1 this series converges absolutely in H(A × G_{p_{0}+1}^{c}). We are done. □ p.3. Proposition 10 enables to obtain some many-dimensional analogue of above cited Polya theorem.

Proposition 11. Let p ≥ 1,

λ_{n} = (λ_{n,1}, λ_{n,2}, …, λ_{n,p}) ∈ ℂ^{p}, n = 1, 2, …

Suppose that the conditions (15) are fulfilled and each sequence { λ_{n,k} }_{n=1}^{∞}, k = 1, …, p, is effective. Let the series (13) converge in some ball K_{0} = { |z − z_{0}|_{p} < δ } to the function f(z). Then f is a single-valued function with the convex domain of existence W_{f}, and the series (13) converges absolutely in H(W_{f}).

Proof. An analytic continuation of f from K_{0} into W_{f} can be realized by the help of every possible chains of polydiscs containing in W_{f} and such that each of them has a nonempty intersection with the following polydisc of the chain. Let K_{1} be a polydisc, for which K_{0} ⋂ K_{1} ≠ ∅ and let f admit an analytic continuation from K_{0} to K_{1}. Let us fix an arbitrary polydisc B_{0} such that B_{0} ⊂ K_{0} ⋂ K_{1}. If we put

×_{j=1}^{p} D_{j} = B_{0}, ×_{j=1}^{p} G_{j} = K_{1} in proposition 10 then by this proposition the series (13)

converges absolutely in H(K_{1}). Lemma 1 implies the absolute convergence of (13) in H((K_{0} ⋃ K_{1})^{c}). If K_{2} is the following polydisc in the chosen chain, then the same arguments show that the series (13) converges absolutely in H((K_{1} ⋃ K_{2})^{c}) and therefore in H(K_{0} ⋃ K_{1} ⋃ K_{2}). Applying the same procedure successively to all polydisc of the chain, we obtain that the series (13) converges absolutely in the union of all polydiscs of the chosen chain.

Since each point z_{1} of W_{f} is contained in some polydisc K from some chain of polydiscs, the series (13) converges absolutely in W_{f}. Hence its sum g is a singlevalued function in W_{f}. Since the series (13) converges absolutely in H(W_{f}), then g ∈ H(W_{f}). It is evident that g(z) gives an analytic extension of f(z) from K_{0} to W_{f}. Due to lemma 1 the series (13) converges absolutely in H(W_{f}^{c}). Hence g ∈ H(W_{f}^{c}), and f can be extended analytically from K_{0} to W_{f}^{c}. Therefore W_{f} = W_{f}^{c}. That completes the proof. □ p.4. Let us apply proposition 10 to one special case. We put G_{j}^{β}(r) = {z_{j} : r < Im z_{j} < β} where r ≥ −∞, r < β ≤ +∞, j = 1, 2, …, p; p ≥ 1;

G^{β}(r) = ×_{j=1}^{p} G_{j}^{β}(r); G_{j}^{β} = G_{j}^{β}(0); G_{0}^{β} = ×_{j=1}^{p} G_{j}^{β}.

Corollary 1. Let the series (13) satisfy condition A) in proposition 10. Suppose that the series (13) converges in the domain G_{0}^{β} and its sum f(z_{1}, …, z_{p}) admits an analytic continuation into the domain G^{β}(−ρ) with some ρ > 0. Then the series (13) converges absolutely in H(G^{β}(−ρ)).

Corollary 2. Let the series (13) satisfy the condition A) and converge in the domain G_{0}^{β}. Let D_{j} be a domain in ℂ such that for each j ≤ p

conv(G_{j}^{β} ⋃ D_{j}) = G_{j}^{β}(−r_{j})

with some r_{j} > 0. Suppose that the sum of the series (13) admits an analytic continuation into the domain

×_{j=1}^{p} (G_{j}^{β} ⋃ D_{j}) .

Then the series (13) converges absolutely in

H(×_{j=1}^{p} G_{j}^{β}(−r_{j})).

For example we can put in corollary 2 D_{j} = { z_{j} : | Im z_{j}| < r_{j} } or D_{j} = K_{j}(r_{j}) := { z_{j} : |z_{j}| < r_{j} }.

Corollary 3. Let the series

∑_{n=1}^{∞} c_{n} exp i (∑_{j=1}^{p} λ_{n,j}z_{j}) , λ_{n,j} ∈ ℝ, c_{n} ∈ ℂ,  (18)

1 ≤ j ≤ p, converge in the domain G_{0}^{β} and let its sum admit an analytic continuation into the domain G^{β}(−ρ) where 0 < ρ ≤ +∞.

Suppose that the exponents λ_{n,j} satisfy the following conditions

lim_{n→∞} \frac{n}{λ_{n,j}} = 0;  j = 1, 2, …, p;  (19)
inf{ |λ_{n,j} − λ_{m,j}| : n ≠ m; n, m = 1, 2, … } > 0,  j = 1, 2, …, p. (20)

Then the series (18) converges absolutely in H(G^{β}(−ρ)).

p.5. Suppose that the series (18) satisfies the condition (15) and, additionally, λ_{n,j} > 0, n ≥ 1, j = 1, 2, . . .. The its maximal domain of convergence coincides with the maximal domain of absolute convergence and must be some tube domain G^{∞}(r). W.l.o.g. one can assume r = 0. Let the series (18) satisfy the conditions (19), (20), and let its sum admit an analytic continuation to some ball K(r) = ×_{j=1}^{p} K_{j}(r). By proposition 10 the series (18) converges absolutely in the domain (G^{∞}(0) ⋃ K(r))^{c}, which is impossible. Hence the boundary of the domain of maximal convergence of the series (18) is a natural boundary for its sum.

This result can be refined. Let the series (18) satisfy the condition A) and converge in the tube domain G_{0}^{∞}. Suppose that the sum of the series (18) admits an analytic continuation to the domain G_{0}^{∞} ⋃ K(r) with some r > 0. According to proposition 10 the series (18) converges absolutely in H((G^{∞} ⋃ K(r))^{c}). We are going to prove that under these assumptions the series (18) converges absolutely in the tube domain ×_{j=1}^{p} G_{j}^{∞}(−r/p). This fact is evident for p = 1, and we again apply the method of induction. We put p = m + 1 and admit that the desired assertion is true for all p ≤ m. Let us fix a point z_{m+1} such that Re z_{m+1} = 0, 0 < Im z_{m+1} < r. It is clear that z_{m+1} ∈ G_{m+1}^{∞} ⋂ K_{m+1}(r). If a_{k} = c_{k} exp(iλ_{k,m+1}z_{m+1}), k ≥ 1, then the series

∑_{k=1}^{∞} a_{k} exp i(∑_{j=1}^{m} λ_{k,j}z_{j})

converges absolutely in the domain H_{m} := ×_{j=1}^{m} G_{j}^{∞}, and its sum admits an analytic continuation into the domain H_{m} ⋃(×_{j=1}^{m} K_{j}(r)). By induction hypothesis this series converges absolutely in the domain ×_{j=1}^{m} G_{j}^{∞}(−r/m). Then every series

∑_{k=1}^{∞} c_{k} exp i[(Re z_{m+1} + i Im z_{m+1})λ_{k,m+1} + ∑_{s=1}^{m} λ_{k,s}z_{s}]

converges absolutely in the domain ×_{j=1}^{m} G_{j}^{∞}(−r/m) for each fixed point z_{m+1} such that −∞ < Re z_{m+1} < +∞, 0 < Im z_{m+1} < r. Hence the initial series (18) converges absolutely in the domain μ_{m+1} := (×_{j=1}^{m} G_{j}^{∞}(−r/m)) × G_{m+1,0}^{r}, where G_{m+1,0}^{r} := { z_{m+1} : 0 < Im z_{m+1} < r }.

Analogously the series (18) converges absolutely in the domain

μ_{l} := G_{l,0}^{r} × ( ×_{{j=1 \atop j≠l}}^{m+1} G_{j}^{∞}(−r/m)), l = 1, 2, …, m.

It should be noted that the multipliers in each Cartesian product μ_{l} are taken in a natural order of succession, so that G_{l,0}^{r} be on the l-th place, G_{1}^{∞}(−r/m) be on the first place, and so on. By lemmas 1, 2 the series (18) converges absolutely in the

domain V = conv( ⋃_{l=1}^{m+1} μ_{l}).

Let us fix θ ∈ (0, 1), ε ∈ (0, θr) and take an arbitrary point w(w_{1}, w_{2}, …, w_{m+1}) from T_{θ} := ×_{j=1}^{m+1} G_{j}^{∞}(−\frac{θr}{m+1}). If ∃l ≤ m + 1: w_{l} ∈ θG_{l,0}^{r}, then w ∈ V_{θ} where

V_{θ} := conv( ⋃_{l=1}^{m+1}(θμ_{l})). Suppose now that for all l ≤ m + 1 w_{l} ∉ θG_{l,0}^{r}. In this case

either −\frac{θr}{m+1} < Im w_{l} ≤ 0 or θr < Im w_{l} < ∞. Let us take (m + 1) points A_{k}: A_{1} = (iε, \frac{m+1}{m}w_{2}, …, \frac{m+1}{m}w_{m+1}), …, A_{m+1} = (\frac{m+1}{m}w_{1}, …, \frac{m+1}{m}w_{m}, iε). It is easy to check that ∀r ≤ m + 1 A_{k} ∈ θμ_{k}. Therefore

\frac{1}{m + 1}( ∑_{k=1}^{m+1} A_{k}) = (\frac{iε}{m + 1} + w_{1}, \frac{iε}{m + 1} + w_{2}, …, \frac{iε}{m + 1} + w_{m+1}) ∈ V_{θ}.

Tending ε to zero we obtain (w_{1}, w_{2}, …, w_{m+1}) ∈ \overline{V}_{θ} ⊂ V. Hence ∀θ ∈ (0, 1)

T_{θ} ⊆ V. Since ×_{j=1}^{m+1} G_{j}^{∞}(−\frac{r}{m+1}) = ⋃_{0<θ<1} T_{θ}, we have ×_{j=1}^{m+1} G_{j}^{∞}(−\frac{r}{m+1}) ⊂ V. So

the series (18) converges absolutely in the domain ×_{j=1}^{m+1} G_{j}^{∞}(−\frac{r}{m+1}), and the proof is finished. We have obtained the following result:

Proposition 12. Let λ_{k,j} ∈ ℝ, j = 1, 2, … m; k = 1, 2, . . .. Let the condition (19) be fulfilled and let each sequence {λ_{k,j}}_{k=1}^{∞}, j = 1, 2, … m, be effective. Suppose that the series (18) converges in the tube domain G_{0}^{∞} = ×_{j=1}^{m} G_{j}^{∞} and its sum f can be extended analytically to the circle K_{r},r > 0. Then the series (18) converges absolutely in H(G^{∞}(−r/m)).

Corollary . Suppose that

λ_{k,j} ∈ ℝ, inf{ |λ_{k,j} − λ_{s,j}| : j = 1, 2, … m; k ≠ s; k, s = 1, 2, … } > 0,

and the condition (19) is fulfilled. If the series (18) converges in G_{0}^{∞} and its sum admits an analytic continuation to K_{r} with r > 0 then the series (18) converges absolutely in H(G^{∞}(−r/m)).

This corollary gives us a more strong result under less restrictive supposition than theorem 2 of Kawai in [1]. p.6.Continuing investigation of the series (18) under assumption A), we suppose now that this series converges in the domain G_{0}^{β} = ×_{j=1}^{p} G_{j}^{β} and its sum admits an analytic continuation into the domain G_{0}^{β} ⋃ K(r), where K(r) = ×_{j=1}^{p} K_{j}(r) and ∞ > r ≥ β > 0. In the case p = 1 above mentioned assumptions imply that the series (18) converges absolutely in the domain G_{1}^{r}(−r) = G^{r}(−r) = { z_{1} : −r < Im z_{1} < r }. We shall prove by induction that in a general situation, for any p ≥ 1, the series (18) converges absolutely in the domain G^{β}(−r/p). We use here just the same induction argument as in the previous case. Taking an arbitrary point z_{m+1} such that Re z_{m+1} = 0, 0 < Im z_{m+1} < β, and applying induction hypothesis for p ≤ m, we obtain that the series (18) converges absolutely in the domain

(×_{j=1}^{p} G_{j}^{β}(−r/m))×G_{m+1}^{β}. Moreover this series converges absolutely in the domain

V := conv( ⋃_{l=1}^{m+1} τ_{l}) with τ_{l} = G_{l}^{β} × [×m+1_{{j=1 \atop j≠l}} G_{j}^{β}(−r/m)], l = 1, 2, …, m + 1. Fixing

arbitrary θ ∈ (0, 1), ε ∈ (0, θβ), and w(w_{1}, …, w_{m+1}) ∈ T_{θ} := ×_{j=1}^{m+1} G_{j}^{θβ}(−\frac{θr}{m+1}) we obtain that if ∃l ≤ n + 1: 0 < Im w_{l} < θβ, then w ∈ θG_{l}^{β} and all the more

w ∈ V_{θ} := conv( ⋃_{l=1}^{m+1} θτ_{l}). If w_{j} ∉ θG_{j}^{β} for all j ≤ m + 1 then −\frac{θr}{m+1} < Im w_{j} ≤ 0,

1 ≤ j ≤ m + 1. Constructing (m + 1) points A_{k} as in p.5 and applying just the same arguments as before, we obtain ultimately that ×_{j=1}^{m+1} G_{j}^{β}(−\frac{r}{m+1}) = ⋃_{0<θ<1} T_{θ} ⊂ V.

p.7.The case r < β < ∞ is more complicated. We restrict ourselves here only with p = 2. Let the series

∑_{k=1}^{∞} c_{k} exp i(λ_{k,1}z_{1} + λ_{k,2}z_{2}),  λ_{k,j} ∈ ℝ; j = 1, 2; k = 1, 2, … (21)

converge in the domain G^{β} := G_{1}^{β} × G_{2}^{β}. As before we assume that the condition A) is fulfilled (for p = 2). Suppose that the sum of the series (21) admits an analytic continuation into the domain G^{β} ⋃ K(r), K(r) = K_{1}(r) × K_{2}(r), and r < β. We fix an arbitrary point z_{2} such that Re z_{2} = 0, 0 < Im z_{2} < r, and put a_{n} = c_{n} exp λ_{n,2}z_{2}, n = 1, 2, . . .. The series

∑_{n=1}^{∞} a_{n} exp iλ_{n,1}z_{1}  (22)

converges absolutely in the domain G_{1}^{β} and its sum has an analytic continuation into the domain G_{1}^{β} ⋃ K_{1}(r). That implies an absolute convergence of the series (22) in the domain G_{1}^{β}(−r). Then the initial series (21) converges absolutely in the domain G_{1}^{β}(−r)×G_{2}^{r}. Analogously the series (21) converges absolutely in the domain G_{1}^{r} × G_{2}^{β}(−r). That implies an absolute convergence of the series (21) in the domain

T := conv[(G_{1}^{r} × G_{2}^{β}(−r)) ⋃(G_{1}^{β}(−r) × G_{2}^{r}) ⋃ G^{β}].

It is easy to check that

T = ℝ^{2}(x) × Γ = { (w_{1}, w_{2}) : −∞ < Re w_{j} < +∞, j = 1, 2; (Im w_{1}, Im w_{2}) ∈ Γ },

where Γ is the interior of the convex polygon Δ in the plane ℝ^{2}(Im w_{1}, Im w_{2}). The vertexes of Δ are seven points (0, β), (β, β), (β, 0), (r, −r), (0, −r), (−r, 0), (−r, r). Simple geometrical arguments show that T contains the domain G_{1}^{(β+r)/2}(−r/2) × G_{2}^{(β+r)/2}(−r/2). If r = β and the same initial suppositions take place, we obtain that the series (21) converges absolutely in the tube domain T_{1} = ℝ^{2}(x) × Γ_{1}, where Γ_{1} is the interior of the polygon Δ_{1} with five vertexes (β, β), (β, −β), (0, −β), (−β, 0), (−β, β). In this case G_{1}^{β}(−β/2) × G_{2}^{β}(−β/2) ⊂ T_{1}.

Finally, if r > β, then the series (21) converges absolutely in the tube domain T_{2} = ℝ^{2}(x) × Γ_{2} where the plane domain Γ_{2} is the interior of the polygon Δ_{2} with eight vertexes (β, r), (r, r), (r, 0), (β, −r), (0, −r), (−r, 0), (−r, β), (0, r). In this situation again G_{1}^{β(−r/2)} × G_{2}^{β(−r/2)} ⊂ T_{2}. It is worth reminding that the absolute convergence of the series (21) in the domain G^{β}(−r/2) also arises from the result obtained in the previous point (p.6) for an arbitrary dimension. p.8. It is worth mentioning that some results of §§2, 4 were published in Russian in the paper [16] hard to come by. Besides,this paper contains some errors and misprints. First of all, according to proposition 5 in [16], if the series (21) satisfies the condition A) with p = 2, converges in the domain G^{β} := G_{1}^{β} × G_{2}^{β}, and its sum has un analytic extension to the domain G^{β} ⋃ K(r), K(r) = K_{1}(r) × K_{2}(r), then the series (21) converges absolutely in the domain

conv[ ⋃_{0<α<1} G_{1}^{β}(−αr) × G_{2}^{β}(−(1 − α)r)].

Unfortunately, the proof of proposition 5 in [16] is based on the false inclusion (G_{1}^{β} ⋃ K_{1}(r)) × G_{2}^{β} ⊂ G^{β} ⋃ K(r) and therefore is incorrect. Probably proposition 5 in [16] as well as its corollary is untrue. Further, there is a “gap” in the proof of proposition 2 in [16] mentioned above (see § 3, p. 4 in this paper). At last the magnitude r cos α at the page 20 of [16] must be replaced by r sin α, and Ω_{l,k}(−δ) at the pages 20, 23—by Ω_{l,k}(−μ) with μ described in the proof of Proposition 1 in [16].

5. The characterization of the complete domain of absolute convergence of many-dimensional Dirichlet series

In this paragraph we investigate the domain of absolute convergence of the series (13) from two points of view. Firstly we fix an arbitrary convex domain G ⊂ ℂ^{p}, p ≥ 1, and determine conditions under which the series (13) converges absolutely in H(G). After that we fix an arbitrary series (13) and characterize the complete domain abc(13) of its absolute convergence

p.1. Let us denote by Concom G the totality of all convex compacts of an arbitrary convex domain G in ℂ^{p}. For each convex set Q ⊂ ℂ^{p} we can introduce its support function H_{Q}(α): H_{Q}(α) = sup{ Re⟨α, z⟩ : z ∈ Q }, α ∈ S^{1} := { w ∈ ℂ^{p} : |w|_{p} = 1 }, where w = (w_{1}, w_{2}, … w_{p}), α = (α_{1}, α_{2}, … α_{p}), z = (z_{1}, z_{2}, … z_{p}) and |z|_{p} =

( ∑_{k=1}^{p} |z_{k}|^{2})^{1/2}. The corollary of lemma 1 gives us the following result.

Proposition 13. If G is an arbitrary convex domain in ℂ^{p}, then the following assertions are equivalent:

1) the series (13) converges absolutely at each point z from G;

2) the series (13) converges absolutely in H(G);

3) ∀Q ∈ Concom G

∑_{n=1}^{∞} |c_{n}| exp |λ_{n}|_{p}H_{Q}(ϕ(λ_{n})) < ∞,

ϕ(λ_{n}) = (ϕ_{1}(λ_{n}), …, ϕ_{p}(λ_{n})), ϕ_{j}(λ_{n}) = \frac{λ_{n,j}}{|λ_{n}|_{p}}, 1 ≤ j ≤ p, λ_{n} = (λ_{n,1}, … λ_{n,p}).

Corollary . If the series (13) converges in H(Q), then ∀Q ∈ Concom G

γ_{Q} := lim sup_{n→∞}(\frac{ln |c_{n}|}{|λ_{n}|_{p}} + H_{Q}(ϕ(λ_{n})) < 0.  (23)

Proof. Suppose that for some Q_{0} ∈ Concom G γ_{Q_{0}} > 0. Then there exists an infinite subsequence of integers n_{k} such that n_{k} ↑ +∞ and ∀k ≥ 1

\frac{ln |c_{n_{k}}|}{|λ_{n_{k}}|_{p}} + H_{Q_{0}}(ϕ(λ_{n_{k}})) > \frac{1}{2}γ_{Q_{0}} =: β.

Hence ∀k ≥ 1

max{ |c_{n_{k}} exp⟨λ_{n_{k}}, z⟩| : z ∈ Q_{0} } = |c_{n_{k}}| exp |λ_{n_{k}}|_{p}H_{Q_{0}}(ϕ(λ_{n_{k}})) > exp |λ_{n_{k}}|_{p}β ≥ 1,

and the series (13) diverges in H(G). We have proved that

∀Q ∈ Concom G γ_{Q} ≤ 0.  (24)

Suppose now that ∃Q_{1} ∈ Concom G: γ_{Q_{1}} = 0. Let us put d = ρ(Q_{1}, ∂G) = max\{ |z − w|_{p} : z ∈ Q_{1}, w ∈ ∂G \}; \overline{K}_{d/2} = \{ z ∈ ℂ_{p} : |z|_{p} ≤ d/2 \}; Q_{d} = Q_{1} + \overline{K}_{d/2} := \{ z_{1} + z_{2} : z_{1} ∈ Q_{1}, z_{2} ∈ K̄_{d/2} \}. It is clear that Q_{d} ∈ Concom G and H_{Q_{d}}(α) = H_{Q_{1}}(α) + \frac{d}{2}, ∀α ∈ S^{1}. Hence

γ_{Q_{d}} = lim sup_{n→∞}[\frac{ln |c_{n}|}{|λ_{n}|_{p}} + H_{Q_{1}}(ϕ(λ_{n})) + \frac{d}{2}] = γ_{Q_{1}} + \frac{d}{2} > 0,

and we came to contradiction with (24). □

Proposition 14. Let the series (13) satisfy conditions (14) and (23). Then this series converges absolutely in H(G).

Proof. Let us fix an arbitrary Q ∈ Concom G. It follows from (23) that there exist B < ∞ and η > 0 such that

∀k ≥ 1 |c_{k}| ≤ B exp(−|λ_{k}|_{p}[H_{Q}(ϕ(λ_{k})) + η]).

Hence for the same k

|c_{k}| max{ exp⟨λ_{k}, z⟩ : z ∈ Q } ≤ B exp(−η|λ_{k}|_{p}) =: δ_{k}

where ∑_{k=1}^{∞} δ_{k} < ∞ by (14). □

Corollary . If the series (13) satisfies the condition (14), then the following assertions are equivalent:

1) the series (13) converges at each point of G ;

2) the series (13) converges absolutely at each point of G ;

3) the series (13) converges in H(G) ;

4) the series (13) converges absolutely in H(G) ;

5) ∀Q ∈ Concom G ∑_{n=1}^{∞} |c_{n}| exp |λ_{n}|_{p}H_{Q}(ϕ(λ_{n})) < ∞;

6) the condition (23) holds.

p.2.Suppose that G is a bounded convex domain in ℂ^{p}. Standard arguments and simple estimates show that the condition (23) is equivalent to the following one:

lim sup_{n→∞}[\frac{ln |c_{n}|}{|λ_{n}|_{p}} + H_{G}(ϕ(λ_{n}))] ≤ 0.  (25)

Thus we obtained the following results.

Proposition 15. If the series (13) converges in a bounded convex domain G then the condition (25) holds.

Proposition 16. If G is a bounded convex domain in ℂ^{p} and the series (13) satisfies (14), then the relation (25) is equivalent to each of assertions 1)–6) in Corollary of Proposition 14.

The conditions of convergence and absolute convergence of the series (13) in a bounded convex domain G ⊂ ℂ^{p} have been obtained by a number of mathematicians. It was proved in [17] for the case p = 1 that if |λ_{n}| → ∞ and the series (13) converges in H(G) where G is a bounded convex domain in ℂ, then the inequality (25) holds. On the other hand, if the series (13) satisfies (14) (again with p = 1), then this series converges absolutely in H(G) ([17], pp. 333–334). Using the same method as in [17], the authors of [18] have proved that if the series (13) converges in H(G), where G = G_{1} × G_{2} × … × G_{p} and G_{j} is a bounded convex domain in ℂ, 1 ≤ j ≤ p, p ≥ 1, and if |λ_{n}|_{p} → ∞, then the condition (25) holds. Besides, if the series (13) satisfies (14) and (25), then the series (13) converges absolutely in H(G), G = G_{1} × G_{2} × … × G_{p} ([18], pp. 460–461).

It is evident that all these results from [17], [18] are contained in Propositions 15–16 and can be partly strengthened. Namely, according to Proposition 15, if the series (13) converges in H(G), then the relation (25) holds. The same assertion was obtained in [17], [18] under additional assumption lim_{n→∞} |λ_{n}|_{p} = ∞.

At last, the recent paper [19] contains the following result.

Theorem 2.1 ([19]). Let G be a bounded convex domain in ℂ^{p}. If the series (13) converges in H(G) and |λ_{n}|_{p} → ∞ as n → ∞, then the condition (25) holds. Conversely, if the series (13) satisfies (14) and (25), then this series converges absolutely in H(G).

It is evident that this assertion is also contained in Proposition 15–16, and the assumption lim_{n→∞} |λ_{n}|_{p} = ∞ in its first part can be omitted. p.3. Let us fix now an arbitrary series (13) and determine a maximal domain D = abc(13) of its absolute convergence. If z_{0} ∈ D then { z_{0} } ∈ Concom D, and by corollary of Proposition 13

lim sup_{n→∞}[\frac{ln |c_{n}|}{|λ_{n}|_{p}} + Re⟨ϕ(λ_{n}), z_{0}⟩] < 0.  (26)

Denote by B_{Λ} the set of all points z_{0}, satisfying (26). It is easy to prove that the set B_{Λ} is convex and open in ℂ^{p}. Besides, D ⊆ B_{Λ}. Let us fix an arbitrary point z_{0} in B_{Λ}. Then ∃h > 0:

lim sup_{n→∞}[\frac{ln |c_{n}|}{|λ_{n}|_{p}} + Re⟨ϕ(λ_{n}), z_{0}⟩] ≤ −2h.

We put V (z_{0}, h) = { z ∈ ℂ^{p} : |z − z_{o}|_{p} < h }. Then ∀z ∈ V (z_{0}, h)

lim sup_{n→∞}[\frac{ln |c_{n}|}{|λ_{n}|_{p}} + h + Re⟨ϕ(λ_{n}), z_{0}⟩] ≤ −h < 0.

If the series (13) satisfies the condition (14), then by Proposition 16 this series converges absolutely in H(V (z_{0}, h)), since ∀α ∈ S^{1} H_{V (z_{0},h)}(α) = h + Re⟨α, z_{0}⟩. We see that the series (13) converges absolutely at each point of B_{Λ}, hence B_{Λ} ⊆ D and, finally, B_{Λ} = D. We have obtained the following result.

Proposition 17. If D is the maximal domain of absolute convergence of the series (13), then D ⊆ B_{Λ}. If, additionally, the condition (14) holds, then D = B_{Λ}.

p.4. In order to determine the support function of the maximal domain D of absolute convergence of the series (13) we apply arguments used previously by G.L. Luntz in one-dimensional case (p = 1) in [20], [21].

If α = (α_{1}, α_{2}, …, α_{p}) ∈ S^{1} and δ > 0 we put

l_{Λ}(α; δ) = lim sup_{n→∞}\{ \frac{ln |c_{n}|}{|λ_{n}|_{p}} : |ϕ_{k}(λ_{n}) − α_{k}| ≤ δ, k = 1, 2, …, p \};
l_{Λ}(α) = lim_{δ↓0} l_{Λ}(α; δ).

If for some α_{0} = (α_{1}^{0}, …, α_{p}^{0}) and δ_{0} > 0 there exists only a finite number of elements λ_{n} such that

|ϕ_{k}(λ_{n}) − α_{k}^{0}| ≤ δ_{0}, k = 1, 2, …, p,

then by definition l_{Λ}(α_{0}) = −∞. It is clear that l_{Λ}(α, δ) ≥ l_{Λ}(α) for all δ > 0 and for all α ∈ S^{1}. We put Λ = { ϕ(λ_{n}) : n = 1, 2, … }. Denote by Λ′ the set of all limit points of Λ.

Proposition 18. The series (13) diverges at each point z_{0} such that ∃α_{0} ∈ S^{1}:

Re ⟨α_{0}, z_{0}⟩ > −l_{Λ}(α_{0}).  (27)

Proof. Inequality (27) implies that l_{Λ}(α_{0}) > −∞. Therefore α_{0} ∈ Λ′. There exist μ > 0, τ > 0 such that

Re ⟨α, z_{0}⟩ > −l_{Λ}(α_{0}) + μ,

if |α − α_{0}|_{p} < τ. Since −l_{Λ}(α_{0}, τ) ≤ −l_{Λ}(α_{0}), we have

Re ⟨α, z_{0}⟩ > −l_{Λ}(α_{0}, τ) + μ,

if |α − α_{0}|_{p} < τ. Suppose at first that l_{Λ}(α_{0}, τ) = +∞. Then

∃\{m_{n}\} : m_{n} ↑ +∞, \frac{ln |c_{m_{n}}|}{|λ_{m_{n}}|_{p}} > n, n = 1, 2, …;

|ϕ_{k}(λ_{m_{n}}) − α_{0,k}| < τ, k = 1, 2, …, p.

Hence

|c_{m_{n}} exp ⟨λ_{m_{n}}, z_{0}⟩ | > exp |λ_{m_{n}}|_{p}(n − |z_{0}|_{p}) > 1 ∀n > N(z_{0}),

and the series (13) diverges at the point z_{0}. Finally, let −∞ < l_{Λ}(α_{0}, τ) < ∞. Then for an arbitrary ε > 0 there exists an infinite sequence of integers (n_{l}) _{l=1}^{+∞} such that

n_{l} ↑ +∞, |c_{n_{l}}| > exp[l_{Λ}(α_{0}, τ) − ε]|λ_{n_{l}}|_{p},
|ϕ_{k}(λ_{n_{l}}) − α_{0,k}| < τ, k = 1, 2, …, p; l = 1, 2, …

Hence

|c_{n_{l}} exp ⟨λ_{n_{l}}, z_{0}⟩ | = |c_{n_{l}}| |exp |λ_{n_{l}}|_{p} ⟨\frac{λ_{n_{l}}}{|λ_{n_{l}}|_{p}} , z_{0}⟩| >
> exp |λ_{n_{l}}|_{p}[l_{Λ}(α_{0}, τ) − ε − l_{Λ}(α_{0}, τ) + μ].

If we take ε = \frac{μ}{2}, then ∀l ≥ 1 |c_{n_{l}}| · | exp ⟨λ_{n_{l}}, z_{0}⟩ | ≥ 1, and the series (13) diverges. □

Let T_{Λ} be the set of all z such that ∀α ∈ Λ′ Re ⟨α, z⟩ < −l_{Λ}(α).

Proposition 19. If condition (14) is fulfilled, then the series (13) converges absolutely at each point of T_{Λ}.

Proof. Let z be an arbitrary fixed point in T_{Λ}. Then ∀α ∈ Λ′ Re ⟨α, z⟩ < −l_{Λ}(α). Let us fix α = (α_{1}, α_{2}, …, a_{p}) ∈ Λ′.

Then ∃h > 0 : Re ⟨α, z⟩ < −l_{Λ}(α) − h. Since

l_{Λ}(α) = lim_{δ↓0} l_{Λ}(α, δ),

there exists δ_{0} > 0 such that ∀δ ∈ (0, δ_{0}) −l_{Λ}(α) < −l_{Λ}(α, δ) + \frac{h}{3}. Let us fix δ ∈ (0, δ_{0}). The function Re ⟨β, z⟩ = ψ(β) being continuous on S^{1}, we can find η > 0 such that

Re ⟨β, z⟩ < −l_{Λ}(α, δ) − \frac{h}{3}

if β = (β_{1}, β_{2}, …, β_{p}) ∈ S^{1} and |β_{k} − α_{k}| < η, k = 1, 2, …, p. We can always assume η < δ. Let us consider the series

∑_{n∈N_{1}} |c_{n}|| exp ⟨λ_{n}, z⟩ |  (28)

where N_{1} = N_{1}(η) is the set of all integers n such that for k = 1, 2, …, p |ϕ_{k}(λ_{n}) − α_{k}| < η. If n ∈ N_{1}(η) and n > N(η, h), then ln |c_{n}| < |λ_{n}|_{p}[l_{Λ}(α, η) + \frac{h}{6}] ≤ |λ_{n}|_{p}[l_{Λ}(α, δ) + \frac{h}{6}]. Hence for the same n

|c_{n}|| exp ⟨λ_{n}, z⟩ | < exp (−\frac{h}{6}|λ_{n}|_{p}) .

By condition (14) the series (28) converges.

Suppose now that α ∉ Λ′. Then for some η_{0} > 0 there exists only a finite number of elements λ_{n} such that |ϕ_{k}(λ_{n}) − α_{k}| ≤ η_{0}. Therefore the series (28) with N_{1} = N_{1}(η_{0}) degenerates to a finite sum.

Thus for each element α ∈ S^{1} there exists its spherical heighborhood v(α): |β_{k} − α_{k}| < η, η = η(α), k = 1, 2, …, p, such that the corresponding series (28) with N_{1} = N_{1}(η(α)) converges. Heine-Borel lemma implies immediately the absolute convergence of the whole series (13). □

Proposition 20. If for some z_{0} ∈ ℂ and α_{0} ∈ Λ′ Re ⟨α_{0}, z_{0}⟩ ≥ −l_{Λ}(α_{0}) then the series (13) cannot converge in any neighbourhood of the point z_{0}.

Proof. Let us fix an arbitrary d > 0 and put

\overline{V} (z_{0}, d) = \{ w : |w − z_{0}|_{p} ≤ d \}.

The point z̃ = (z̃_{1}, …, z̃_{p}) with

z̃_{k} = z_{0,k} + d exp(−i arg α_{0,k}), k = 1, 2, …, p,

belongs to \overline{V} (z_{0}, d), and

Re ⟨α_{0}, z̃⟩ = Re ⟨α_{0}, z_{0}⟩ + d ∑_{k=1}^{p} |α_{0,k}| > l_{Λ}(α_{0}).

Therefore the series (13) diverges at the point z̃. □

Let us denote by M_{Λ} convex closed set

M_{Λ} := ⋂_{|α|_{p}=1} {z ∈ ℂ^{p} : Re ⟨α, z⟩ ≤ −l(α)} .

It is evident that z ∉ M_{Λ} iff ∃α_{0} ∈ S^{1} : Re ⟨a_{0}, z⟩ > −l(α_{0}). Hence by proposition 18 the series (13) diverges at each point of ℂ \ M_{Λ}. Further it is easy to check that int M_{Λ} = T_{Λ}. If the series (13) satisfies the condition (14) then the maximal domain abc(13) of its absolute convergence coincides with convex domain B_{Λ} = T_{Λ} = int M_{Λ} with the support function H_{M_{Λ}}(α). This function satisfies the inequality H_{M_{Λ}}(α) ≤ −l(α), ∀α ∈ S^{1}.

The paper [22] contains assertion (p. 69) according to which if series (13) satisfies (14), then B_{Λ} = int M_{Λ} = abc(13). Besides, the series (13) diverges at each point exterior to M_{Λ}. The author of [22] communicates (without proof) that these results can be obtained by method used in one-dimensional situation in [20]. For the reader’s convenience we gave here a detailed proof of these results with some refinements and additions (for example, proposition 19 connected with the set Λ′ which has not been considered in the paper [22]). p.5. Let the series (13) with condition (14) converge in some convex domain G ⊂ ℂ^{p} with the support function H_{G}(α). Then G ⊆ abc(13) = int M_{Λ} = T_{Λ}. Hence ∀α ∈ S^{1} H_{G}(α) ≤ −l_{Λ}(α). The latter inequality is equivalent to the following one:

∀α ∈ Λ′ H_{G}(α) ≤ −l_{Λ}(α).

We put for any α from S^{1}

G(α) := { z ∈ ℂ^{p} : Re ⟨α, z⟩ < H_{G}(α)} .

Since Re ⟨α, z⟩ < H_{G}(α), ∀α ∈ S^{1}, ∀z ∈ G and

T_{Λ} = { z ∈ ℂ^{p} : Re ⟨α, z⟩ < −l_{Λ}(α), ∀α ∈ S^{1} }
= { z ∈ ℂ^{p} : Re ⟨α, z⟩ < −l(α), ∀α ∈ Λ′ } ,

we have G ⊆ G[Λ′] := ⋂_{α∈Λ′} G_{Λ}(α) ⊆ T_{Λ} = abc(13). Following [23] we call G[Λ′] the

polar enveloping of the domain G with respect to Λ′. The results obtained above lead directly to the following analogue of results on analytic extension of solutions of homogeneous convolution equations in ℂ^{p} (cf. Corollary 3.2 of [24] and Theorem 3.3 in [23]).

Proposition 21. Let the series (13) converge in some convex domain G ⊂ ℂ^{p} and let the condition (14) be satisfied. Then the series (13) converges absolutely in the polar enveloping G[Λ′] of G with respect to Λ′.

6. Another form of many-dimensional Dirichlet series

Of concern here is the many-dimensional Dirichlet series

∑_{∥(n)∥_{p}=1}^{∞} c_{(n)} exp ⟨λ_{(n)}, z⟩  (29)

where (n) = (n_{1}, n_{2}, …, n_{p}), p ≥ 1; n_{j} = 1, 2, …; j = 1, 2, …, p; λ = (λ_{1}, λ_{2}, …,

λ_{p}), ∥λ∥_{p} = ∑_{k=1}^{p} |λ_{k}|, λ_{(n)} = (λ_{1,n_{1}}, λ_{2,n_{2}}, …, λ_{p,n_{p}}); Λ = (λ_{(n)})_{|(n)|_{p}=p}^{∞}, and λ_{(n)} are

pairwise different, i.e. if (n) ≠ (m) then ∃j ≤ p : λ_{j,n_{j}} ≠ λ_{j,m_{j}}. The properties of the series (29) are very similar to those of the series (13), that is the series

∑_{n=1}^{∞} c_{n} exp ⟨λ_{n}, z⟩  (30)

with λ_{n} = (λ_{n,1}, λ_{n,2}, …, λ_{n,p}), n ≥ 1. Here we describe briefly the properties of the series (29) using the well known inequalities: ∀λ ∈ ℂ^{p}, ∀p ≥ 1 |λ|_{p} ≤ ∥λ∥_{p} ≤ \sqrt{p}|λ|_{p}.

Lemma 3. The set Q of all points of absolute convergence of the series (29) is convex. The series (29) converges absolutely in H(int Q).

The proof is a literal repetition of the proof of lemma 1 with the change c_{n} by c_{(n)} and λ_{n,j} by λ_{j,n_{j}}. Strictly speaking we need to prove only an absolute convergence of (29) in H(int Q) since the set Q of all points of absolute convergence of the series (29) does not depend on the order of summation.

Corollary 1. If G is an arbitrary convex domain in ℂ^{∣} then the following assertions are equivalent:

1) the series (29) converges absolutely in the H(G);

2) the series (29) converges absolutely at each point of G.

Corollary 2. If the series (29) converges absolutely in a neighborhood of some point z_{0} from ℂ^{∣} then there exists the maximal domain abc(29) of absolute convergence of the series (29). This domain is convex and coincides with int Q where Q is the set from lemma 1.

Lemma 4. Let the series (29) satisfy the condition

lim_{∥(n)∥_{p}→∞} \frac{ln ∥(n)∥_{p}}{∥λ_{(n)}∥_{p}} = 0.  (31)

Then the maximal domain of convergence of the series (29) coincides with the maximal domain of its absolute convergence.

The proof of this lemma also repeats literally the proof of lemma 2 with the same change c_{n} by c_{(n)} and λ_{n} by λ_{(n)}. The concluding inequality in the proof is the following one

∀(n) |c_{(n)}|| exp ⟨λ_{(n)}, z_{0}⟩ | ≤ M exp (−\frac{δ}{2\sqrt{p}}∥λ_{(n)}∥_{p}) =: h_{(n)}

with ∑_{∥(n)∥_{p}=p}^{∞} h_{(n)} < ∞ and z_{0} = (z_{1,0}, …, z_{p,0}).

It is evident that the equality (31) is equivalent to the following one:

lim_{|(n)|_{p}→∞} \frac{ln |(n)|_{p}}{|λ_{(n)}|_{p}} = 0.

Corollary . Let G be an arbitrary convex domain in ℂ^{∣} and let the equality (31) hold. Then each assertion 1),2) in Corollary 1 of lemma 3 is equivalent to every one from the following two

3) the series (29) converges absolutely at each point of G;

4) the series (29) converges in H(G).

The following results give us the conditions of absolute convergence of the series (29) in some convex domain G. They are very similar to those stated in §5 for the series (30).

Corollary 1 of lemma 1 leads directly to

Proposition 22. If G is an arbitrary convex domain in ℂ^{∣}, then the following assertions are equivalent :

1) the series (29) converges absolutely at each point of domain G;

2) the series (29) converges absolutely in H(G);

3) ∀Q ∈ Concom G

∑_{∥(n)∥_{p}=p}^{∞} |c_{(n)}| exp |λ_{(n)}|_{p}H_{Q}(ψ(λ_{(n)})) < ∞,

where ψ(λ_{(n)}) = (ψ_{1}(λ_{(n)}), …, ψ_{p}(λ_{(n)})), ψ_{j}(λ_{(n)}) = \frac{λ_{n,j}}{|λ_{(n)}|_{p}}, 1 ≤ j ≤ p, H_{Q}(α) is

the support function of the convex set Q : H_{Q}(α) = sup{Re ⟨α, z⟩ : z ∈ Q}, α ∈ S^{1} := { w ∈ ℂ^{∣} : |≾|_{∣} = ⊮ } and Concom G is the totality of all convex compacts of the domain G.

Corollary . If the series (29) converges absolutely in H(G), then ∀Q ∈ Concom G

γ_{Q} := lim sup_{∥(n)∥_{p}→∞} (\frac{ln |c_{(n)}|}{|λ_{(n)}|_{p}} + H_{Q}(ψ(λ_{(n)}))) < 0.  (32)

Proof. [The proof of corollary] Let us assume that γ_{Q_{0}} > 0 for some Q_{0} from Concom G. Then there exists a sequence (n_{k}) = (n_{k,1}, n_{k,2}, …, n_{k,p}) such that ∥(n_{k})∥_{p} ↑ +∞ as k → ∞ and

∀k ≥ 1 \frac{ln |c_{(n_{k})}|}{|λ_{(n_{k})}|_{p}} + H_{Q}(ψ(λ_{(n_{k})})) ≥ \frac{γ_{Q_{0}}}{2} =: β.

Then ∀k ≥ 1

max{|c_{(n_{k})} exp⟨λ_{(n_{k})}, z⟩| : z ∈ Q_{0}} = |c_{(n_{k})}| exp[|λ_{(n_{k})}|_{p}H_{Q_{0}}(ψ(λ_{(n_{k})}))] >

> exp β|λ_{(n_{k})}|_{p} ≥ 1, (33)

and the series (29) diverges in H(G), which contradicts to the initial supposition. So

∀Q ∈ Concom G  γ_{Q} ≤ ′.  (34)

Let us assume now that γ_{Q_{1}} = 0 for some Q_{1} from Concom G. We put as above d = ρ(Q_{1}, ∂G) = max\{|‡ − ⊒|_{√} : ‡ ∈ Q_{∞}, ⊒ ∈ ∂G\}, \overline{K_{∇}} = \{z ∈ ℂ^{∣} : |𝟋|_{∣} ≤ ∖\},

Q_{d} = Q_{1} + \overline{K_{\frac{d}{2}}} .

Then Q_{d} ∈ Concom G, and H_{Q_{d}}(α) = H_{Q_{1}}(α) + \frac{d}{2}, ∀α ∈ S^{1}. Hence

γ_{Q_{d}} = lim sup_{∥(n)∥_{p}→∞} [\frac{ln |c_{(n)}|}{|λ_{(n)}|_{p}} + H_{Q_{1}}(ψ(λ_{(n)})) + \frac{d}{2}] = γ_{Q_{1}} + \frac{d}{2} > 0,

which contradicts to (34). □

Proposition 23. If the series (29) satisfies the conditions (31) and (32), then this series converges absolutely in H(G).

Proof. Let us fix an arbitrary Q from Concom G. The condition (32) implies

∃η > 0 ∃B < ∞ ∀(n) |c_{(n)}| ≤ B exp [−|λ_{(n)}|_{p}(H_{Q}(ψ(λ_{(n)})) + η)] .

Hence for the same (n)

|c_{(n)}| max{| exp ⟨λ_{(n)}, z⟩ | : z ∈ Q} ≤ B exp(−η|λ_{(n)}|_{p}) =: δ_{(n)},

where ∑_{∥(n)∥_{p}=p}^{∞} δ_{(n)} < ∞ by virtue of (31). □

Corollary . If the series (29) satisfies (31), then the following assertions are equivalent:

1) the series (29) converges at each point of G;

2) the series (29) converges in H(G);

3) the series (29) converges absolutely at each point of G;

4) the series (29) converges absolutely in H(G);

5) ∀Q ∈ Concom G ∑_{∥(n)∥_{p}=p}^{∞} |c_{(n)}| exp |λ_{(n)}|_{p}H_{Q}(ψ(λ_{(n)})) < +∞;

6) the condition (32) holds.

If the convex domain G is bounded, the condition (32) is equivalent to the following one

lim sup_{∥(n)∥_{p}→∞} [\frac{ln |c_{(n)}|}{|λ_{(n)}|_{p}} + H_{G}(ψ(λ_{(n)}))] ≤ 0.  (35)

It is clear that the relation (35) is in its turn equivalent to such inequality

lim sup_{|(n)|_{p}→∞} [\frac{ln |c_{(n)}|}{|λ_{(n)}|_{p}} + H_{G}(ψ(λ_{(n)}))] ≤ 0.

Taking into account the corollary of Proposition 22 we obtain the following result which does not suppose the validity of (31).

Proposition 24. If the series (29) converges in some bounded convex domain G, then the condition (35) holds.

Further, the following result is valid

Proposition 25. If G is a bounded convex domain in ℂ^{∣} and the series (29) satisfies the condition (31) then the relation (35) is equivalent to each of assertions 1)–6) in Corollary of Proposition 23.

It has been proved previously in the paper [25] that under the condition (31) the assertion 4) is equivalent to (35), if G is a bounded convex domain in ℂ^{∣}.

Let us characterize now the maximal domain D = abc(29) of absolute convergence of the series (29), as it was done for the series (30) in p.3, §5. Let B_{Λ} be the set of all points z_{0} such that

lim sup_{∥(n)∥_{p}→∞} [\frac{ln |c_{(n)}|}{|λ_{(n)}|_{p}} + Re ⟨ψ(λ_{(n)}), z_{0}⟩] < 0.  (36)

If z_{0} ∈ D then {z_{0}} ∈ Concom D and due to Corollary of Proposition 22 the inequality (36) holds. So D ⊆ B_{Λ}. It is evident that B_{Λ} is a convex and open. In just the same way as in p.3 §5, and only with the insignificant change of designations we show that if (31) holds then B_{Λ} ⊆ D. Finally, B_{Λ} = D and the following result is valid.

Proposition 26. Always D ⊆ B_{Λ}, and if (31) holds then D = B_{Λ}.

In order to find the support function of the domain D we use as in p.4 §5 the techniques applied previously in one-dimensional case by G.L.Luntz. For δ > 0 and α = (α_{1}, …, α_{p}) ∈ S^{1} we put

l_{Λ}(α; δ) := lim sup_{∥(n)∥_{p}→∞} \{\frac{ln |c_{(n)}|}{|λ_{(n)}|_{p}} : |ψ_{k}(λ_{(n)}) − α_{k}| ≤ δ, k = 1, 2, …, p\};

l_{Λ}(α) := lim_{δ→0} l_{Λ}(α; δ). As in [1], if for some δ_{0} > 0 and α^{0} ∈ S^{1} there exists only a

finite number of elements λ_{(n)} such that |ψ_{k}(λ_{(n)}) − α_{k}^{0}| ≤ δ_{0}, k = 1, 2, …, p, we put l_{Λ}(α^{0}) = −∞.

The following result is proved just in the same way as Proposition 18.

Proposition 27. The series (29) diverges at each point z_{0} such that ∃ α^{0} ∈ S^{1} :

Re ⟨α^{0}, z_{0}⟩ > −l_{Λ}(α^{0}).  (37)

Let Λ′, as above, be the totality of all partial limits (limit points) of the sequence Λ = {λ_{(n)}}, and let T_{Λ} be the set of all z such that ∀α ∈ Λ′ Re ⟨α, z⟩ < −l_{Λ}(α).

Proposition 28. If the condition (31) is valid, then the series (29) converges absolutely at each point of T_{Λ}.

This result is proved in the same way as Proposition 19.

Proposition 29. If for some z_{0} ∈ ℂ^{∣}, α^{⊬} ∈ Λ′ we have Re ⟨α^{0}, z_{0}⟩ ≥ −l_{Λ}(α^{0}), then the series (29) can not converge at any neighborhood of the point z_{0}.

The proof of this proposition does not differ practically from the proof of Proposition 20.

Denote by M_{Λ} the convex closed set M_{Λ} := ⋂_{|α|_{p}=1}{ z ∈ ℂ^{∣} : Re ⟨α, 𝟋⟩ ≤ −⋖_{Λ}(α) }.

Then (as it has been shown in the end of p.4 §5) the series (29) diverges at each point of ℂ \ 𝕄_{Λ}, and int M_{Λ} = T_{Λ}. Further, if condition (31) holds, then the domain D = abc(29) coincides with the convex domain B_{Λ} = T_{Λ} = int M_{Λ}, and H_{M_{Λ}}(α) ≤ −l_{Λ}(α), ∀α ∈ S^{1}.

Let G be a convex domain in ℂ^{∣} with the support function H_{G}(α). As in the end of §5 we put G[Λ′] := ⋂_{α∈Λ′} G(α) where α ∈ S^{1} and G(α) := { ‡ ∈ ℂ^{∣} : Re ⟨α, 𝟋⟩ <

ℍ_{G}(α) }. The set G[Λ′] is the polar enveloping of the domain G with respect to Λ′ [23].

If the series (29) converges in G and the condition (31) holds, then G ⊆ ⊣⌊⌋(∈∃) = int M_{Λ}, and (see the end of §5) ∀α ∈ S^{1} H_{G}(α) ≤ −l_{Λ}(α). Hence we obtain: G ⊆ G[Λ′] ⊆ T_{Λ} = ⊣⌊⌋(∈∃). So the following result is valid.

Proposition 30. If the series (29) converges in some convex domain G ⊂ ℂ^{∣} and the condition (31) holds, then the series (29) converges absolutely in G[Λ′].

7. Concluding remarks

The results proved in §§2–5 are relative to the following problems. Suppose that the series (13) satisfies the condition (14) and converges in a neighborhood of some point z_{0} ∈ ℂ^{p}, p ≥ 1. Then the maximal domain of convergence of the series (13) coincides with the maximal domain abc(13) of its absolute convergence.

In this situation two problems arise naturally:

1) effective characterization of abc(13) in terms connected with c_{n} and λ_{n};

2) determination of additional assumptions under which the domain abc(13) coincides with the complete domain W_{f} of the sum f of the series (13).

The results obtained in §§2–5 enable to formulate the summarizing assertion which decides entirely the first problem and partially the second one.

Proposition 31. Suppose that the series (13) converges in some neighborhood of the point z_{0} ∈ ℂ^{p}, p ≥ 1. Suppose also that at least one of the following three assumptions takes place:

1) each sequence {λ_{n,j}}_{n=1}^{∞} is effective, j = 1, 2, …, p, and the equality (14) holds;

2) relations (19) are valid and lim_{δ→0} lim sup_{k→∞} \frac{1}{|λ_{k,j}|} ∫_{0}^{δ|λ_{k,j}|} \frac{n(λ_{k,j},t)−1}{t}dt ≤ 0, j =

1, 2, …, p;

3) conditions (19) and (20) are fulfilled. The abc(13) = W_{f} = T_{Λ} = B_{Λ} = int M_{Λ}.

It is worth mentioning that 3) ⇒ 2) ⇒ 1) and each assumption in this chain can be checked more easily than the successive one.

In conclusion the author would like to set up some problems connected directly with subjects of this paper.

Problem 1. It would be very interesting to obtain a many-dimensional analogue of Polya theorem and in particular to characterize the domain of holomorphy of the sum of this series. Proposition 31 gives us a similar and even a more strong result but under one of additional assumptions. It is not clear whether the exact manydimensional analogue of Polya theorem is valid without these additional assumptions and only under condition (19).

Problem 2. We say that the sequence { λ^{(n)} }_{n=1}^{∞},

λ^{(n)} = (λ_{n,1}, …, λ_{n,p}), n = 1, 2, …

with lim sup n → ∞\frac{n}{λ_{n,j}} = 0 is effective in ℂ^{p}, if all series (13) with the exponents

λ^{(n)} have the following property. Suppose that the series (13) converges in some domain Q ⊂ ℂ^{p} and the sum of this series can be extended analytically to some domain G ⊃ Q. Then the series (13) converges in G. In other words the sequence { λ^{(n)} }_{n=1}^{∞} is said to be effective if any series (13) with the exponents { λ^{(n)} }_{n=1}^{∞} converges in the complete domain of holomorphy of its sum.

It follows from propositions 10–11 that if each sequence

{ λ_{n,k} }_{n=1}^{∞}, k = 1, 2, …, p

is effective in ℂ and conditions (15) hold, then {λ^{(n)}}_{n=1}^{∞} is effective in ℂ^{p}. It is unknown whether the inverse assertion is valid. Besides, we do not know a criterion for the sequence {λ^{(n)}}_{n=1}^{∞} be effective in ℂ^{p}, p ≥ 1.

Problem 3. It is evident that the maximal domain T of absolute convergence of the series (18) is the tube domain R^{p}(x) × B where B is some domain in ℝ which is

Yu.F. Korobeĭnik 43

convex due to lemma 1:

T = { (x_{k} + iy_{k})_{k=1}^{p} : −∞ < x_{k} < +∞, (y_{1}, y_{2}, …, y_{p}) ∈ B } .

However it seems to be unknown whether the domain of holomorphy of the sum f of the series (18) is also some tube domain. In solving this problem of characterization of the domain of holomorphy of the sum of the series (18) one may impose on the exponents λ^{(n)} additional restrictions such as (15) or (19).

Problem 4. As we have seen for ourselves, a great many results proved in §§4-5 for many-dimensional series (30) can be transferred without any difficulties to the series (29). However, the problem of extension of the initial domain of convergence of the series (29) to the domain of analytic continuation of its sum has turned out to be essentially more difficult. In particular, it is not known yet whether an analogue of Proposition 10 for the series (29) is valid. Consequently, we have not succeeded in obtaining a many-dimensional generalization of Polya theorem similar to Propositions 11 and 31 for the series (29).

The paper has been partly supported by Russian Fund of Fundamental Investigations.

References

[1] Kawai T. The Fabry-Ehrenpreis gap theorem and linear differential operators of infinite order // Journal of Mathematics, 1987, v. 109, p. 57–64.

[2] Leont’ev A.F. Rjadi exponent (Series of exponentials).—Moscow: Nauka, 1976, 536p. (Russian).

[3] Leont’ev A.F. Rjadi polynomov Dirichlet i ich obobschenija (Series of Dirichlet polynomials and their generalizations). Moscow, Trudi Matem. Instit. im. V.A.Steklova, 1951 , V. 39 (Russian).

[4] Korobeinik Yu.F. On a dual problem. II. Applications to LN^{∗}-spaces and other questions // Math. USSR Sbornik, 1975, v. 27, 1, p. 1–22; transl. into English from Matem. Sbornik, 1975, t. 98(140), 1, p. 3–26.

[5] Bratischev A.V. On interpolation problem in some classes of entire functions // Sibirskii Mat. Journal. 1976. v. XVII, 1, p. 30–43; Engl. translation in Sib. Mathem. Journal. 1976.

[6] Bratischev A.V. Sufficient conditions for interpolating sets in classes of entire functions characterized by the indicator // Dokl. Acad. Nauk SSSR. 1984. v. 279, 5, p. 1036–1039; Engl. transl. in Soviet Math. Dokl., 1984.

[7] Grishin A.F., Russakovski A.M. A free interpolation by entire functions // Teorija functij, functional’nij analiz i ich prilozhenija (Theory of functions, functional analysis and their applications). Kharkov. 1985. v. 44, p. 32–44. (Russian).

[8] Hille E. Note on Dirichlet series with complex exponents // Ann. of Mathem. 1924. vol.25, p.251–278.

[9] Polya G. Eine Verallegemeinerung des Fabryschen Lückensatzes // Nachr. Gessel. Wiss. Göttingen. 1927. p.187-195.

[10] Levin B.Ja. Distribution of Zeros of Entire Functions. Translations of Mathem. Monografs. 1980. vol. 5. AMS Providence, Rhode Island.

[11] Bernstein V. Lecons sur le progress recents de la theorie séries de Dirichlet.—Paris, 1933.

[12] Valiron G. Sur l’abscisse de convergence des séries de Dirichlet // Bull. Soc. Math. de France. 1924. v. 52.

[13] Havin V.P. Prostranstva analyticheskih functij (The spases of analytic functions) // In Coll. ”Itogi nauki. Ser. mat. anal.” Moscow, VINITI, 1964, p. 76–164. (Russian).

[14] Kantorovich L.V. and Akilov G.P. Functional analysis in normed spases. Moscow. Fizmatgiz. 1959. (Russian).

[15] Valiron G. Sur les solutions des equations différentielles linéaires d’ordre infini et à coefficients constants // Ann. scient. École norm. supér. 1929. V. 46. 1, p. 25–53.

[16] Korobeinik Yu.F. K voprosu ob analiticheskom prodolzhenii rjadov Dirichlet (On analytic continuation of Dirichlet series) // Matematychne Studii., Lviv, 1995, 4, p. 18–19. (Russian).

[17] Korobeinik Yu.F. Representing systems // Izvestija Acad. Nauk SSSR. Ser. Matem. 1978. vol. 42, 2, p. 325-355; English transl. in USSR Mathem. Izvestija, 1978.

44 ABSOLUTELY CONVERGENT DIRICHLET SERIES

[18] Le Khai Khoi and Yu.F. Korobeinik. Representing systems of exponential functions in polycylindric domains // Mat. Sbornik. 1983, vol.122(164), 4, p. 458–473; English transl. in Math. USSR Sborn. 1985, vol. 50, 2.

[19] Le Khai Khoi. Holomorphic Dirichlet Series in several variables // Mathematica Scandinavica. 1995. vol.1977, Fasc. 1, p. 85–107.

[20] Luntz G. L. O nekotorikh obobschennikh rjadakh Dirichlet (On some generalized Dirichlet series) // Matem. Sbornik. 1942. vol. 52, 1, p. 33-50. (Russian).

[21] Luntz G. L. Ob odnom klasse obobschennikh rjadov Dirichlet (On one class of generalized Dirichlet series) // Uspekhi Mat. Nauk. 1957, t. XII, 3, p. 173-179. (Russian).

[22] Gabovich Z.G. Teoremi tipa Ostrovskogo dlja functii nescol’kikh peremennikh, predstavimikh rjadami Dirichlet (Ostrovsky-type theorems for functions of several variables represented by Dirichlet series) // In Coll. ”Teorija functij , functional’nij” analis i ich prilozhenija” (”Theory of functions, functional analysis and their applications”). 1972. vol. 16, p. 68–73. Kharkov University Press (Russian).

[23] Aoki T. Existence and continuation of holomorphic solutions of differential equations of infinite order // Adv. in Math. 1988. vol. 72, p. 261–283.

[24] Sebbar A. Prolongement des solutions holomorphes de certain operateurs différentiels d’ordre infini à coefficient constants. In Lecture Notes in Mathematics. 1980. vol. 822, p. 199–220. Springer Verlag. New York/Berlin.

[25] Gromov V.P. Criteria on convergence of exponential series in functional spaces // Dokl. Akad. Nauk SSSR. 1991, vol. 318, 1, p. 20-23; English transl. in Soviet Mat. Dokl. 1991.

Faculty of Mechanics and Mathematics, Rostov State University, 5 Zorge st., Rostov on Don, 344090 Russia WWW Home Page: http://www.unird.ac.ru/mexmat/ma/kor.en.html E-mail address: kor@mmf.unird.ac.ru

Received November 4, 1997