PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 77(91) (2005), 71–78

ON THE DISTRIBUTION OF M-TUPLES OF B-NUMBERS

Werner Georg Nowak

Communicated by Aleksandar Ivić

Abstract. In the classical sense, the set B consists of all integers which can be written as a sum of two perfect squares. In other words, these are the values attained by norms of integral ideals over the Gaussian field ℚ(i). G. J. Rieger (1965) and T. Cochrane, R. E. Dressler (1987) established bounds for the number of pairs (n, n + h), resp., triples (n, n + 1, n + 2) of B-numbers up to a large real parameter x. The present article generalizes these investigations into two directions: The result obtained deals with arbitrary M-tuples of arithmetic progressions of positive integers, excluding the trivial case that one of them is a constant multiple of one of the others. Furthermore, the estimate applies to the case of an arbitrary normal extension K of the rational field instead of ℚ(i).

1. Introduction. Already E. Landau’s in his classic monograph [4] provided a proof of the result that the set B of all positive integers which can be written as a sum of two squares of integers is distributed fairly regularly: It satisfies the asymptotic formula

(1.1)  ∑_{1⩽n⩽x, n∈B}1 ∼ \frac{c x}{\sqrt{log x}}  (c > 0) .

Almost six decades later, G. J. Rieger [9] was the first to deal with the question of “B-twins”: How frequently does it happen that both n and n + 1 belong to the

2000 Mathematics Subject Classification. Primary 11P05, 11N35. Key words and phrases. B-numbers; Selberg sieve; norms of ideals in number fields. The author gratefully acknowledges support from the Austrian Science Fund (FWF) under

project Nr. P18079-N12.

set B? A bit more general, he was able to show that, for any positive integer h and large real x,

(1.2)  ∑_{{1⩽n⩽x \atop n∈B, n+h∈B}}1 ≪ ∏_{{p|h \atop p≡3 (mod 4)}} (1 + \frac{1}{p}) \frac{x}{log x} .

Later on, C. Hooley [2] and K.-H. Indlekofer [3], independently and at about the same time, showed that this bound is essentially best possible. In 1987, T. Cochrane and R. E. Dressler [1] extended the question to triples of B-numbers. Replacing Rieger’s sieve technique by a more recent variant of Selberg’s method, they succeeded in proving that

(1.3)  ∑_{{1⩽n⩽x \atop n∈B, n+1∈B, n+2∈B}}1 ≪ \frac{x}{(log x)^{3/2}} .

2. Statement of result. In this article we intend to generalize these estimates in two different directions: Firstly, instead of pairs or triples we consider M-tuples of arithmetic progressions (a_{m} n + b_{m}), m = 1, …, M ⩾ 2, where a_{m} ∈ ℤ^{+}, b_{m} ∈ ℤ throughout. Secondly, we deal with an arbitrary number field K which is supposed to be a normal extension of the rationals of degree [K : ℚ] = N ⩾ 2. Denoting by O_{K} the ring of algebraic integers in K, we put

b_{K}(n) := \{ \begin{matrix}1 & if there exists an integral ideal 𝔄 in O_{K} of norm N (𝔄) = n, \\ 0 & else. \end{matrix}

Our target is then the estimation of the sum

(2.1)  S(x) = S(a_{1}, b_{1}, …, a_{M}, b_{M}; x) := ∑_{1⩽n⩽x} ∏_{m=1}^{M} b_{K}(a_{m} n + b_{m}) .

Of course, the classic case reported in section 1 is contained in this, by the special choice K = ℚ(i), the Gaussian field.

Theorem. Suppose that (a_{m}, b_{m}) ∈ ℤ^{+} × ℤ for m = 1, …, M, and, furthermore,

∏_{{m,k=1 \atop m≠k}}^{M}(a_{m}b_{k} − a_{k}b_{m}) ≠ 0 .

Then, for large real x,

S(a_{1}, b_{1}, …, a_{M}, b_{M}; x) ≪ γ(a_{1}, b_{1}, …, a_{M}, b_{M}) \frac{x}{(log x)^{M(1−1/N)}} ,

with

γ(a_{1}, b_{1}, …, a_{M}, b_{M}) = ∏_{p∈ℙ′} (1 + \frac{M}{p} ) ,

the finite set of primes ℙ′ = ℙ′(a_{1}, b_{1}, …, a_{M}, b_{M}) to be defined below in (4.6). The ≪-constant depends on M and the field K, but not on a_{1}, b_{1}, …, a_{M}, b_{M}.

3. Some auxiliary results. Notation. Variables of summation automatically range over all integers satisfying the conditions indicated. p denotes rational primes throughout, and ℙ is the set of all rational primes. 𝔓 stands for prime ideals in O_{K}. For any subset ℙ^{∘} ⊆ ℙ, we denote by D(ℙ^{∘}) the set of all positive integers whose prime divisors all belong to ℙ^{∘}. The constants implied in the symbols O(·), ≪, ≫, etc., may depend throughout on the field K and on M, but not on a_{1}, b_{1}, …, a_{M}, b_{M}.

Lemma 1. For each prime power p^{α}, α ⩾ 1, let \overline{Ω}(p^{α}) be a set of distinct residue classes \overline{𝔠} modulo p^{α}. Define further

Ω(p^{α}) = \{n ∈ ℤ^{+} : n ∈ ⋃_{\overline{𝔠}∈\overline{Ω}(p^{α})} \overline{𝔠}\} ,

and let

θ(p^{α}) := 1 − ∑_{j=1}^{α} \frac{#\overline{Ω}(p^{j})}{p^{j}} > 0 , θ(1) := 1 .

Suppose that Ω(p^{α}) ∩ Ω(p^{α′}) = ∅ for all primes p and positive integers α ≠ α′. For real x > 0, let finally

A(x) = {n ∈ ℤ^{+} : n ⩽ x & n ∉ ⋃_{p∈ℙ, α∈ℤ^{+}}Ω(p^{α})} .

Then, for arbitrary real Y > 1,

#A(x) ⩽ \frac{x + Y^{2}}{V_{Y}} ,

where

V_{Y} := ∑_{0<d<Y} ∏_{p^{α} ∥ d} (\frac{1}{θ(p^{α})} − \frac{1}{θ(p^{α−1})}) .

Proof. This is a deep sieve theorem due to A. Selberg [10]. It can be found in Y. Motohashi [5, p. 11], and also in T. Cochrane and R. E. Dressler [1].

Lemma 2. Let (c_{n})_{n∈ℤ^{+}} be a sequence of nonnegative reals, and suppose that the Dirichlet series

f(s) = ∑_{n=1}^{∞} c_{n} n^{−s}

converges for Re(s) > 1. Assume further that, for some real constants A and β > 0,

f(s) = (A + o(1))(s − 1)^{−β} ,

as s → 1+. Then, for x → ∞,

∑_{1⩽n⩽x} \frac{c_{n}}{n} = (\frac{A}{Γ(1 + β)} + o(1))(log x)^{β} .

Proof. This is a standard Tauberian theorem. For the present formulation, cf. Cochrane and Dressler [1, Lemma B].

4. Proof of the Theorem. We recall the decomposition laws in a normal extension K over ℚ of degree N ⩾ 2 (cf. W. Narkiewicz [6, Theorem 7.10]): Every rational prime p which does not divide the field discriminant disc(K) belongs to one of the classes

ℙ_{r} = {p ∈ ℙ : (p) = 𝔓_{1} ⋯ 𝔓_{N/r}, N (𝔓_{1}) = ⋯ = N (𝔓_{N/r}) = p^{r}} ,

where r ranges over the divisors of N, and 𝔓_{1}, …, 𝔓_{N/r} are distinct. As an easy consequence, if p ∈ ℙ_{r}, α ∈ ℤ^{+},

(4.1)  b_{K}(p^{α}) = \{ \begin{matrix}1, & if r|α, \\ 0, & else. \end{matrix}

In order to apply Lemma 1, we need a bit of preparation. Let

ℙ_{r}^{∗} = \{p ∈ ℙ_{r} : p ∤ ∏_{m=1}^{M} a_{m} ∏_{{m,k=1 \atop m≠k}}^{M}(a_{m}b_{k} − a_{k}b_{m}), p ≠ M − 1\} ,

ℙ^{∗} = ⋃_{r|N, r>1} ℙ_{r}^{∗} .

Then we choose

\overline{Ω}(p^{α}) := ⋃_{m=1}^{M} \{\overline{a_{m}} ^{(−1)}\overline{(jp^{α−1} − b_{m})} : j = 1, …, p − 1\} ,

if p ∈ ℙ_{r}^{∗} and r ∤ (α−1), while \overline{Ω}(p^{α}) := ∅ in all other cases. Here \overline{·} denotes residue classes modulo p^{α}, in particular \overline{a_{m}} ^{(−1)} is the class which satisfies \overline{a_{m}} \overline{a_{m}} ^{(−1)} = ¯1 mod p^{α}. We summarize the relevant properties of these sets \overline{Ω}(p^{α}), and of the corresponding sets Ω(p^{α}) (see Lemma 1), as follows.

Proposition. Suppose throughout that p ∈ ℙ^{∗} and α ∈ ℤ^{+}.

(i) If p ∈ ℙ_{r}^{∗}, r ∤ (α − 1), then \overline{Ω}(p^{α}) contains exactly M(p − 1) elements. (ii) If a positive integer k lies in some Ω(p^{α}), it follows that there exists an

m ∈ {1, …, M} such that p^{α−1} ∥ (a_{m} k + b_{m}).

(iii) It is impossible that there exist m, n ∈ {1, …, M}, m ≠ n, and a positive integer k, such that some p ∈ ℙ^{∗} divides both a_{m}k + b_{m} and a_{n}k + b_{n}.

(iv) If k ∈ Ω(p^{α}), it follows that

p^{α−1} ∥ ∏_{m=1}^{M}(a_{m} k + b_{m}) .

Consequently, Ω(p^{α}) ∩ Ω(p^{α′}) = ∅ for any positive integers α ≠ α′.

(v) If k ∈ Ω(p^{α}), then

∏_{m=1}^{M} b_{K}(a_{m} k + b_{m}) = 0 .

As a consequence,

S(x) ⩽ #A(x) ,

where S(x) and A(x) have been defined in (2.1) and Lemma 1, respectively.

Proof of the Proposition. (i) Assume that two of these residue classes would be equal, say, \overline{a_{m}} ^{(−1)}\overline{(u p^{α−1} − b_{m})} and \overline{a_{n}} ^{(−1)}\overline{(v p^{α−1} − b_{n})}, where u, v ∈ \{1, …, p − 1\}, m, n ∈ \{1, …, M\}. Multiplying by \overline{a_{m}} \overline{a_{n}}, we could conclude that

a_{n}(u p^{α−1} − b_{m}) ≡ a_{m}(v p^{α−1} − b_{n}) mod p^{α} ,

or, equivalently, that

(4.2)  (a_{n} u − a_{m} v)p^{α−1} ≡ a_{n} b_{m} − a_{m} b_{n}  mod p^{α} .

Hence p |(a_{n} b_{m} − a_{m} b_{n}), which is only possible if m = n. This in turn simplifies (4.2) to

a_{m}(u − v)p^{α−1} ≡ 0 mod p^{α} ,

thus also u = v. □

(ii) If k ∈ Ω(p^{α}), there exist j ∈ {1, …, p − 1}, m ∈ {1, …, M}, and an integer q, such that

a_{m} k = j p^{α−1} − b_{m} + q p^{α} .

From this the assertion is obvious. □

(iii) Assuming the contrary, we would infer that p divides

(a_{m} k + b_{m})b_{n} − (a_{n} k + b_{n})b_{m} = (a_{m} b_{n} − a_{n} b_{m})k ,

hence p |k, thus p divides also b_{m} and b_{n}, which contradicts p ∈ ℙ^{∗}. □

(iv) This is immediate from (ii) and (iii). □

(v) By (ii), p^{α−1} ∥ (a_{m} k + b_{m}) for some m ∈ {1, …, M}. Recalling that r ∤ (α−1) (otherwise Ω(p^{α}) would be empty), along with (4.1) and the multiplicativity of b_{K}(·), it is clear that b_{K}(a_{m}k + b_{m}) = 0. The last inequality is obvious from the relevant definitions. □

We are now ready to apply Lemma 1. Choosing Y = \sqrt{x} and appealing to part (v) of the Proposition, we see that

(4.3)  S(x) ⩽ \frac{2x}{V_{Y}} .

To derive a lower bound for V_{Y}, observe that \overline{Ω}(p) = ∅ for every prime p, and #\overline{Ω}(p^{2}) = M(p−1) for each p ∈ ℙ^{∗}. Further, if p ∉ ℙ^{∗}, then \overline{Ω}(p^{j}) = ∅ throughout. Therefore, if p ∈ ℙ^{∗}, then θ(p) = 1 and

θ(p^{2}) = 1 − \frac{1}{p^{2}} #\overline{Ω}(p^{2}) = 1 − \frac{M(p − 1)}{p^{2}} ,

hence

(4.4)  \frac{1}{θ(p^{2})} − \frac{1}{θ(p)} = \frac{M(p − 1)}{p^{2} − M(p − 1)} ⩾ \frac{M}{p} .

Furthermore, for any α > 2,

θ(p^{α}) ⩾ 1 − M(p − 1) ∑_{2⩽j⩽α} p^{−j} > 1 − \frac{M}{p} ⩾ 0 ,

since M(p − 1) ⩽ p^{2} − 1 according to clause (i) of the Proposition, and M = p + 1 is impossible for p ∈ ℙ^{∗}. Thus actually θ(p^{α}) > 0 for all primes p and all α ∈ ℤ^{+}. Thus all the terms in the sum V_{Y} are nonnegative, and restricting the summation to the set

Q := {d = d_{1}^{2} : d_{1} ∈ ℤ^{+} , μ(d_{1}) ≠ 0 , d_{1} ∈ D(ℙ^{∗})} ,

we conclude by (4.4) that

(4.5) \begin{aligned}V_{Y} & ⩾ ∑_{0<d<Y, d∈Q} ∏_{p|d} (\frac{1}{θ(p^{2})} − \frac{1}{θ(p)}) ⩾ ∑_{0<d_{1}<\sqrt{Y} , d_{1}∈D(ℙ^{∗})} μ^{2}(d_{1}) ∏_{p|d_{1}} \frac{M}{p} = \\ & = ∑_{0<d_{1}<\sqrt{Y} , d_{1}∈D(ℙ^{∗})} μ^{2}(d_{1}) \frac{M^{ω(d_{1})}}{d_{1}} , \end{aligned}

where ω(d_{1}) denotes the number of primes dividing d_{1}. Our next step is to take care of the primes excluded in the construction of ℙ^{∗}. We define (4.6)

ℙ′ = ℙ′(a_{1}, b_{1}, …, a_{M}, b_{M}) := \{p ∈ ⋃_{{r|N \atop r>1}} ℙ_{r} : p | ∏_{m=1}^{M} a_{m} ∏_{{m,k=1 \atop m≠k}}^{M}(a_{m}b_{k} − a_{k}b_{m})\}

and

(4.7) γ = γ(a_{1}, b_{1}, …, a_{M}, b_{M}) := ∏_{p∈ℙ′} (1 + \frac{M}{p} ) = ∑_{k_{1}∈D(ℙ′)} μ^{2}(k_{1}) \frac{M^{ω(k_{1})}}{k_{1}} .

Putting finally

ℙ_{∪} := ⋃_{r|N, r>1} ℙ_{r} ,

we readily infer from (4.5) and (4.7) that

(4.8)  γ V_{Y} ≫ ∑_{k<\sqrt{Y} , k∈D(ℙ_{∪})} μ^{2}(k) \frac{M^{ω(k)}}{k} .

We shall estimate this latter sum by the corresponding generating function

f(s) := ∑_{k∈D(ℙ_{∪})} μ^{2}(k) \frac{M^{ω(k)}}{k^{s}} = ∏_{p∈ℙ_{∪}} (1 + \frac{M}{p^{s}} ) (Re(s) > 1) ,

applying Lemma 2. By h_{1}(s), h_{2}(s), … we will denote functions which are holomorphic and bounded, both from above and away from zero, in every half-plane Re(s) ⩾ σ_{0} > 1/2. We first observe that

(4.9)  f(s) = h_{1}(s) ∏_{p∈ℙ_{∪}}(1 − p^{−s})^{−M}  (Re(s) > 1) .

This follows by a standard argument which can be found exposed neatly in G. Tenenbaum [11, p. 200]. The next step is to consider the Euler product of the Dedekind zeta-function ζ_{K}(s): For Re(s) > 1,

ζ_{K}(s) = ∏_{𝔓} (1 − N (𝔓)^{−s})^{−1} = h_{2}(s) ∏_{r|N} ( ∏_{p∈ℙ_{r}}(1 − p^{−rs})^{−N/r}) =

= h_{3}(s) ∏_{p∈ℙ_{1}}(1 − p^{−s})^{−N} .

Therefore,

\frac{(ζ(s))^{M}}{(ζ_{K}(s))^{M/N}} = h_{4}(s) ∏_{p∈ℙ_{∪}}(1 − p^{−s})^{−M} (Re(s) > 1) .

Comparing this with (4.9), we arrive at

f(s) = h_{5}(s) \frac{(ζ(s))^{M}}{(ζ_{K}(s))^{M/N}} .

From this it is evident that, as s → 1+,

f(s) ∼ h_{5}(1) ρ_{K}^{−M/N} (s − 1)^{−M+M/N} ,

where ρ_{K} denotes the residue of ζ_{K}(s) at s = 1. Lemma 2 now immediately implies that

∑_{k<\sqrt{Y} , k∈D(ℙ_{∪})} μ^{2}(k) \frac{M^{ω(k)}}{k} ≫ (log Y)^{M−M/N} ≫ (log x)^{M−M/N} ,

in view of our earlier choice Y = \sqrt{x}. Combing this with (4.3) and (4.8), we complete the proof of our Theorem. □

5. Concluding remarks. 1. Taking more care and imposing special conditions on the numbers a_{1}, b_{1}, …, a_{M}, b_{M}, one could improve slightly on the factor γ in our estimate. (Observe that Rieger’s bound (1.2) is in fact a bit sharper than our general result.) But it is easy to see that γ is rather small anyway: By elementary facts about the Euler totient function (see K. Prachar [8, p. 24–28]),

γ(a_{1}, b_{1}, …, a_{M}, b_{M}) ≪ ∏_{p∈ℙ′} (1 − \frac{1}{p})^{−M} ≪ (log log x)^{M} ,

under the very mild restriction that, for some constant c > 0,

max_{m=1,...,M}(a_{m}, |b_{m}|) ≪ exp((log x)^{c}) .

2. As far as the asymptotics (1.1) is concerned, the generalization to an arbitrary normal extension K of ℚ can be found in W. Narkiewicz’ monograph [6, p. 361, Prop. 7.11], where it is attributed to E. Wirsing. For this question, the case of non-normal extensions K has been dealt with by R. W. K. Odoni [7]. It may be interesting to extend our present problem to the non-normal case as well. We might return to this at a later occasion.

References

[1] T. Cochrane and R. E. Dressler, Consecutive triples of sums of two squares, Arch. Math. (Basel) 49 (1987), 301–304.

[2] C. Hooley, On the intervals between numbers that are sums of two squares, III, J. Reine Angew. Math. 267 (1974), 207–218.

[3] K.-H. Indlekofer, Scharfe untere Abschätzung für die Anzahlfunktion der B-Zwillinge, Acta Arith. 26 (1974), 207–212.

[4] E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Teubner, Leipzig, 1909.

[5] Y. Motohashi, Lectures on Sieve Methods and Prime Number Theory, Tata Institute of Fund. Research, Bombay, 1983.

[6] W. Narkiewicz, Elementary and analytic theory of algebraic numbers, 2nd ed., Springer & PWN, Berlin, 1990.

[7] R. W. K. Odoni, On the norms of algebraic integers, Mathematika 22 (1975), 71–80.

[8] K. Prachar, Primzahlverteilung, Springer, Berlin–Göttingen–Heidelberg, 1957.

[9] G. J. Rieger, Aufeinanderfolgende Zahlen als Summen von zwei Quadraten, Indag. Math. 27 (1965), 208–220.

[10] A. Selberg, Remarks on multiplicative functions, Number Theory Day, Proc. Conf., New York, 1976, Lecture Notes in Math. 626 (1977), 232–241.

[11] G. Tenenbaum, Introduction to analytic and probabilistic number theory, Cambridge University Press, 1995.

Institute of Mathematics (Received 05 10 2004) Department of Integrative Biology Universität für Bodenkultur Wien Gregor Mendel-Straße 33 1180 Wien, Austria

nowak@mail.boku.ac.at
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