Mean Values of a Class of Arithmetical Functions

Deyu Zhang^{1}

School of Mathematical Sciences Shandong Normal University

Jinan 250014 Shandong P. R. China

zdy 78@yahoo.com.cn

Wenguang Zhai^{1} Department of Mathematics

China University of Mining and Technology Beijing, 100083 P. R. China

zhaiwg@hotmail.com

Abstract

In this paper we consider a class of functions U of arithmetical functions which include P̃(n)/n, where P̃(n) := n ∏_{p|n}(2 − \frac{1}{p}). For any given U ∈ U, we obtain the asymptotic formula for ∑_{n≤x} U(n), which improves a result of De Koninck and Kátai.

1 Introduction

In 1933, Pillai [10] introduced the function

P (n) = ∑_{k=1}^{n} gcd(k, n),

1This work is supported by National Natural Science Foundation of China(Grant Nos. 10771127, 11001154) and Shandong Province Natural Science Foundation (Nos. BS2009SF018, ZR2010AQ009).

and proved that

P (n) = ∑_{d|n} dφ(n/d), and ∑_{d|n} P (d) = nd(n) = ∑_{d|n} σ(d)φ(n/d),

where φ is Euler’s function, d(n) and σ(n) denote the number of divisors of n and the sum of the divisors of n respectively. Many authors investigated the properties of P (n), see [2, 3, 4, 5, 6, 10, 13]; it is Sloane’s sequence \underline{A018804}. Chidambaraswamy and Sitaramachandrarao [6] showed that, given an arbitrary ϵ > 0,

∑_{n≤x} P (n) = e_{1}x^{2} log x + e_{2}x^{2} + O(x^{1+θ+ϵ}),

where e_{1}, e_{2} are computable constants and 0 < θ < 1/2 is some exponent contained in

∑_{n≤x} d(n) = x log x + (2γ − 1)x + O(x^{θ+ϵ}).  (1)

The asymptotic formula (1) is the well-known Dirichlet divisor problem. The latest value of θ is θ = 131/416 proved by Huxley [8].

Tóth [12] first defined the gcd-sum function over regular integers modulo n by the relation

P̃(n) = ∑_{k∈Reg_{n}} gcd(k, n),  (2)

where Reg_{n} = {k : 1 ≤ k ≤ n and k is regular (mod n)}, and proved that P̃(n) is multiplicative and for every n ≥ 1,

P̃(n) = n ∏_{p|n}(2 − \frac{1}{p}).  (3)

It is sequence \underline{A176345} in Sloane’s Encyclopedia. He also obtained the following asymptotic formula

∑_{n≤x} P̃(n) = \frac{x^{2}}{2ζ(2)}(K_{1} log x + K_{2}) + O(x^{3/2}δ(x)),  (4)

where K_{1} and K_{2} are certain constants and δ(x) is given by

δ(x) = exp(−A(log x)^{3/5}(log log x)^{−1/5}).

Zhang and Zhai [15] showed that the estimate of ∑_{n≤x} P̃(n) is closely related to the squarefree divisor problem and improved the error term of (4) under RH.

De Koninck and Kátai [7] introduced two wide classes of arithmetical functions R and U, the first of which includes the function P (n)/n, and the second of which includes P̃(n)/n. More precisely, the class R is made of the following functions R. Firstly let γ(n) denote the kernel of n ≥ 2, that is γ(n) = ∏_{p|n} p (with γ(1) = 1). Then, given an arbitrary positive constant c, an arbitrary real number α > 0 and a multiplicative function κ(n) satisfying | κ(n) |≤ \frac{c}{γ(n)^{α}} for all n ≥ 2, let R ∈ R be defined by

R(n) = R_{κ,c,α}(n) := d(n) ∑_{d∥n} κ(d) = d(n) ∏_{p^{a}∥n}(1 + κ(p^{a})).  (5)

It is easily seen that if we let κ(p^{a}) = −\frac{a/(a+1)}{p}, then the corresponding function R(n) is precisely P (n)/n.

De Koninck and Kátai [7] showed that

T (x) := ∑_{n≤x} R(n) = A_{0}x log x + B_{0}x + O(x^{β+ϵ}),  (6)

with

β = \begin{cases}θ, & if α ≥ 1 − θ; \\ 1 − α, & if α < 1 − θ; \end{cases}

where θ is the exponent in (1), A_{0}, B_{0} are certain constants.

As for the class of functions U, it is made of the functions

U(n) = U_{h,c,α}(n) := 2^{ω(n)} ∑_{d|n} h(d),

where ω(n) stands for the number of distinct prime factors of n, and h is a multiplicative function satisfying |h(n)| ≤ \frac{c}{γ(n)^{α}} for all n ≥ 2. It is easily seen that by taking h(p) = −\frac{1}{2p} and h(p^{a}) = 0, for a ≥ 2, we obtain the particular case U(n) = P̃(n)/n. De Koninck and Kátai [7] proved that

S(x) := ∑_{n≤x} U(n) = t_{1}x log x + t_{2}x + O(\frac{x}{log x}),  (7)

where t_{1}, t_{2} are certain constants.

In this paper, we shall prove the following

Theorem 1. Suppose 0 ≤ α < 1. Then we have

S(x) = t_{1}x log x + t_{2}x + O(x^{1−α+ϵ} + x^{1/2+ϵ}).  (8)

Remark 2. (i) From our proof we see that the evaluation of S(x) is closely related to the distribution of the zeros of the Riemann zeta function. The exponent 1/2 can be reduced to 4/11 if RH is true.

(ii) The exponent 1−α in the error term of Theorem 1 is best possible when α is small. For example, if we take h(n) = n^{−α} with 0 < α < 1/2, then our proof with slight modifications yields

∑_{n≤x} U(n) = t_{1}x log x + t_{2}x + t_{3}x^{1−α} log x + t_{4}x^{1−α} + O(x^{1/2+ϵ}).

We are also interested in the short interval case. In this case, the restrictions on α and RH can be removed. Actually, we have the following Theorem 3. Theorem 3. Suppose (1) holds for 1/4 < θ < 1/3. Then for x^{θ+2ϵ} ≤ y ≤ x, we have

∑_{x<n≤x+y} U(n) = H(x + y) − H(x) + O(yx^{− \frac{ϵ}{2}} + x^{θ+ϵ}),  (9)

where H(x) = t_{1}x log x + t_{2}x.

2 Preliminary Lemmas

Lemma 4. Let s be a complex number with ℜs > 1. Then

∑_{n=1}^{∞} \frac{U(n)}{n^{s}} = \frac{ζ^{2}(s)}{ζ(2s)}G(s),

where G(s) can be written as a Dirichlet series G(s) = ∑_{n=1}^{∞} \frac{g(n)}{n^{s}}, which is absolutely convergent for ℜs > 1 − α. Moreover g(n) satisfies |g(n)| ≪ n^{−α+ϵ}. Proof. For ℜs > 1, by Euler product representation we have

F (s) := ∑_{n=1}^{∞} \frac{U(n)}{n^{s}} = ∏_{p} (1 + ∑_{β=1}^{∞} \frac{U(p^{β})}{p^{βs}}) ,

where U(p^{β}) = 2(1 + h(p) + ⋯ + h(p^{β})), β ≥ 1. Thus

1 + ∑_{β=1}^{∞} \frac{U(p^{β})}{p^{βs}} = 1 + ∑_{β=1}^{∞} \frac{2}{p^{βs}} + 2 ∑_{β=1}^{∞} p^{−βs} ∑_{j=1}^{β} h(p^{j})

= \frac{1 − p^{−2s}}{(1 − p^{−s})^{2}} + 2 ∑_{β=1}^{∞} p^{−βs} ∑_{j=1}^{β} h(p^{j})

= \frac{1 − p^{−2s}}{(1 − p^{−s})^{2}} × (1 + \frac{2(1 − p^{−s})^{2}}{1 − p^{−2s}} ∑_{β=1}^{∞} p^{−βs} ∑_{j=1}^{β} h(p^{j})) ,

hence we get

∑_{n=1}^{∞} \frac{U(n)}{n^{s}} = \frac{ζ^{2}(s)}{ζ(2s)}G(s),

where

G(s) = ∏_{p} (1 + \frac{2(1 − p^{−s})^{2}}{1 − p^{−2s}} ∑_{β=1}^{∞} p^{−βs} ∑_{j=1}^{β} h(p^{j})) .

From the above formula, it is easy to see that G(s) can be expanded to a Dirichlet series G(s) = ∑_{n=1}^{∞} \frac{g(n)}{n^{s}}, which is absolutely convergent for ℜs > 1 − α, if we notice that |h(p)| ≤ \frac{c}{p^{α}}. Therefore |g(n)| ≪ n^{−α+ϵ}.

Lemma 5. Let

∑_{n=1}^{∞} \frac{d^{(2)}(n)}{n^{s}} = \frac{ζ^{2}(s)}{ζ(2s)}, ℜs > 1,

where d^{(2)}(n) denote the number of square-free divisors of n. Then for any real numbers x ≥ 1, we have

D^{(2)}(x) := ∑_{n≤x} d^{(2)}(n) = c_{1}x log x + c_{2}x + Δ^{(2)}(x)

with Δ^{(2)}(x) = O(x^{1/2} log x), where

c_{1} = \frac{1}{ζ(2)}, c_{2} = \frac{2γ − 1}{ζ(2)} − \frac{2ζ′(2)}{ζ^{2}(2)}.

Moreover, if RH is true, then Δ^{(2)}(x) = O(x^{4/11+ϵ}). Proof. The first result is due to Mertens [9] and the second one is due to Baker [1]. Lemma 6.

∑_{n≤x} |g(n)| ≪ x^{1−α+ϵ}.

Proof. It follows from |g(n)| ≪ n^{−α+ϵ}. Lemma 7. Let k ≥ 2 be a fixed integer , 1 < y ≤ x be large real numbers and

A(x, y; k, ϵ) := ∑_{m>x^{ϵ}^{x<nm^{k}≤x+y}} 1.

Then we have

A(x, y; k, ϵ) ≪ yx^{−ϵ} + x^{1/4}.

Proof. This is Lemma 3 of Zhai [14].

3 Proof of Theorem 1

Notice that

\frac{ζ^{2}(s)}{ζ(2s)} = ∑_{ℓ=1}^{∞} \frac{d^{(2)}(ℓ)}{ℓ^{s}}, G(s) = ∑_{m=1}^{∞} \frac{g(m)}{m^{s}}.  (10)

By the Dirichlet convolution, we have

∑_{n≤x} U(n) = ∑_{mℓ≤x} g(m)d^{(2)}(ℓ) = ∑_{m≤x} g(m) ∑_{ℓ≤x/m} d^{(2)}(l),

and Lemma 5 applied to the inner sum gives

∑_{n≤x} U(n) = ∑_{m≤x} g(m) \{\frac{c_{1}x}{m} log(\frac{x}{m}) + \frac{c_{2}x}{m} + O ((\frac{x}{m})^{1/2+ϵ})\}

= c_{1}x \{(log x + \frac{c_{2}}{c_{1}} ) ∑_{m≤x} \frac{g(m)}{m} − ∑_{m≤x} \frac{g(m) log m}{m}\} + O (x^{1/2+ϵ} ∑_{m≤x} \frac{|g(m)|}{m^{1/2+ϵ}} )

= c_{1}x \{(log x + \frac{c_{2}}{c_{1}} ) ∑_{m=1}^{∞} \frac{g(m)}{m} − ∑_{m=1}^{∞} \frac{g(m) log m}{m} + O(x^{−α+ϵ})\} + O (x^{1/2+ϵ} ∑_{m≤x} \frac{|g(m)|}{m^{1/2+ϵ}} ) ,

if we notice by Lemma 6 that both of the infinite series ∑_{m=1}^{∞} \frac{g(m)}{m}, ∑_{m=1}^{∞} \frac{g(m) log m}{m} are absolutely convergent, and

∑_{m>x} \frac{g(m)}{m} ≪ x^{−α+ϵ}, ∑_{m>x} \frac{g(m) log m}{m} ≪ x^{−α+ϵ}.  (11)

Then we have

∑_{n≤x} U(n) = t_{1}x log x + t_{2}x + O(x^{1−α+ϵ}) + O (x^{1/2+ϵ} ∑_{m≤x} \frac{|g(m)|}{m^{1/2+ϵ}} ) ,  (12)

where

t_{1} = \frac{1}{ζ(2)} ∑_{m=1}^{∞} \frac{g(m)}{m} = \frac{G(1)}{ζ(2)},

t_{2} = \frac{1}{ζ(2)} \{(2γ − 1 − \frac{2ζ′(2)}{ζ(2)}) ∑_{m=1}^{∞} \frac{g(m)}{m} − ∑_{m=1}^{∞} \frac{g(m) log m}{m}\}

= \frac{1}{ζ(2)} \{(2γ − 1 − \frac{2ζ′(2)}{ζ(2)})G(1) − G′(1)\} .

By Lemma 6, we have

∑_{m≤x} \frac{|g(m)|}{m^{1/2+ϵ}} ≤ ∑_{m≤x} \frac{1}{m^{1/2+α+ϵ}} ≤ \begin{cases}x^{ϵ}, & α ≥ 1/2; \\ x^{1/2−α+ϵ}, & α < 1/2, \end{cases}

Theorem 1 follows from the above estimates and Eq. (12).

4 Proof of Theorem 3

By Lemma 4, we have

U(n) = ∑_{n=n_{1}n_{2}n_{3}^{2}} d(n_{1})g(n_{2})μ(n_{3}),

where d(n) is the divisor function. Then

∑_{x<n≤x+y} U(n) = ∑_{x<n_{1}n_{2}n_{3}^{2}≤x+y} d(n_{1})g(n_{2})μ(n_{3}) = Σ_{1} + O (Σ_{2} + Σ_{3}) ,  (13)

where

Σ_{1} = ∑_{{n_{2}≤x^{ϵ} \atop n_{3}≤x^{ϵ}}} g(n_{2})μ(n_{3}) ∑_{_{\frac{x}{n_{2}n^{2}_{3}}} <n_{1}≤_{\frac{x+y}{n_{2}n^{2}_{3}}}} d(n_{1}),

Σ_{2} = ∑_{{x<n_{1}n_{2}n^{2}_{3}≤x+y \atop n_{2}>x^{ϵ}}} d(n_{1})|g(n_{2})|,

Σ_{3} = ∑_{{x<n_{1}n_{2}n^{2}_{3}≤x+y \atop n_{3}>x^{ϵ}}} d(n_{1})|g(n_{2})|.

Recalling (1), the inner sum in Σ_{1} is

\frac{(x + y)}{n_{2}n_{3}^{2}} log \frac{(x + y)}{n_{2}n_{3}^{2}} − \frac{x}{n_{2}n_{3}^{2}} log \frac{x}{n_{2}n_{3}^{2}} + (2γ − 1)\frac{y}{n_{2}n_{3}^{2}} + O (\frac{x^{θ}}{n_{2}^{θ}n_{3}^{2θ}} )

= \frac{(x + y) log(x + y) − x log x}{n_{2}n_{3}^{2}} − y \frac{log(n_{2}n_{3}^{2})}{n_{2}n_{3}^{2}} + (2γ − 1)\frac{y}{n_{2}n_{3}^{2}} + O (\frac{x^{θ}}{n_{2}^{θ}n_{3}^{2θ}} ) .

Inserting the above expression into Σ_{1} and after some easy calculations, we get

Σ_{1} = H(x + y) − H(x) + O (yx^{−ϵ} + y^{−αϵ+ϵ^{2}} + x^{θ+ϵ}) .  (14)

For Σ_{2}, we have

|g(n_{2})| ≪ n_{2}^{−α+ϵ} ≪ x^{−αϵ+ϵ^{2}},

if we notice that n_{2} > x^{ϵ}, and hence

Σ_{2} ≪ x^{−αϵ+ϵ^{2}} ∑_{x<n_{1}n_{2}n_{3}^{2}≤x+y} d(n_{1}) = x^{−αϵ+ϵ^{2}} ∑_{x<n≤x+y} d_{∗}(n),

where

d_{∗}(n) = ∑_{n=n_{1}n_{2}n_{3}^{2}} d(n_{1}) ≪ n^{ϵ^{2}}.

Therefore we have

Σ_{2} ≪ x^{−αϵ+ϵ^{2}} ∑_{x<n≤x+y} n^{ϵ^{2}} ≪ yx^{−αϵ+ϵ^{2}}.  (15)

Since d(n) ≪ n^{ϵ^{2}}, g(n_{2}) ≪ 1, by Lemma 7 we have

Σ_{3} ≪ x^{ϵ^{2}} ∑_{{x<n_{1}n_{2}n^{2}_{3}≤x+y \atop n_{3}>x^{ϵ}}} 1 ≪ x^{ϵ^{2}} ∑_{{x<nn^{2}_{3}≤x+y \atop n_{3}>x^{ϵ}}} d(n)

≪ x^{2ϵ^{2}} ∑_{{x<nn^{2}_{3}≤x+y \atop n_{3}>x^{ϵ}}} 1 = x^{2ϵ^{2}}A(x, y; 2, ϵ)

≪ yx^{−ϵ+2ϵ^{2}} + x^{1/4+ϵ^{2}}.  (16)

Then Theorem 3 follows from Eqs. (13)–(16).

5 Acknowledgments

The authors express their gratitude to the referee for a careful reading of the manuscript and many valuable suggestions, which highly improve the quality of this paper.

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2010 Mathematics Subject Classification: Primary 11N37. Keywords: gcd-sum function, regular integers modulo n, Riemann hypothesis, short interval result.

(Concerned with sequences \underline{A018804} and \underline{A176345}.)

Received January 25 2011; revised version received May 24 2011. Published in Journal of Integer Sequences, June 10 2011.

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