Ramanujan expansions of arithmetic functions of several variables

TL;DR

This paper extends recent work on Ramanujan expansions to functions of multiple variables, demonstrating parallels between classical and unitary Ramanujan sums in these expansions.

Contribution

It generalizes existing results on Ramanujan expansions from two-variable functions to functions of several variables, highlighting similarities in properties across different types of sums.

Findings

Extended Ramanujan expansion results to multiple variables

Identified parallels between classical and unitary Ramanujan sums

Provided theoretical framework for multi-variable arithmetic functions

Abstract

We generalize certain recent results of Ushiroya concerning Ramanujan expansions of arithmetic functions of two variables. We also show that some properties on expansions of arithmetic functions of one and several variables using classical and unitary Ramanujan sums, respectively, run parallel.

Equations

$$\sum _{n=1}^{\infty }{2}^{\omega (n)}\frac{|(\mu \ast f)(n)|}{n}<\infty .$$

$$f(n)=\sum _{q=1}^{\infty }{a}_q{c}_q(n),$$

$$a_q=\sum _{m=1}^{\infty }\frac{(\mu \ast f)(mq)}{mq}(q\in \text{mathds}N).$$

$$M(f)=\underset{x\to \infty }{\lim }\frac{1}{x}\sum _{n\le x}f(n)$$

$$\frac{\sigma ((n_1,n_2))}{(n_1,n_2)}=\zeta (3)\sum _{q_1,q_2=1}^{\infty }\frac{c_{q_1}(n_1)c_{q_2}(n_2)}{[q_1,q_2{]}^3},$$

$$\tau ((n_1,n_2))=\zeta (2)\sum _{q_1,q_2=1}^{\infty }\frac{c_{q_1}(n_1)c_{q_2}(n_2)}{[q_1,q_2{]}^2}.$$

$$\frac{\sigma (n)}{n}=\zeta (2)\sum _{q=1}^{\infty }\frac{c_q(n)}{q^2}(n\in \text{mathds}N),$$

$$\tau (n)=-\sum _{q=1}^{\infty }\frac{\log q}{q}{c}_q(n)(n\in \text{mathds}N)$$

$$\frac{\sigma (n)}{n}=\zeta (2)\sum _{q=1}^{\infty }\frac{{\varphi }_2(q)c_q^{\ast }(n)}{q^4}(n\in \text{mathds}N),$$

$$\frac{\sigma ((n_1,n_2))}{(n_1,n_2)}=\zeta (3)\sum _{q_1,q_2=1}^{\infty }\frac{{\varphi }_3([q_1,q_2])c_{q_1}^{\ast }(n_1)c_{q_2}^{\ast }(n_2)}{[q_1,q_2{]}^6}(n_1,n_2\in \text{mathds}N),$$

$$\tau ((n_1,n_2))=\zeta (2)\sum _{q_1,q_2=1}^{\infty }\frac{{\varphi }_2([q_1,q_2])c_{q_1}^{\ast }(n_1)c_{q_2}^{\ast }(n_2)}{[q_1,q_2{]}^4}(n_1,n_2\in \text{mathds}N),$$

$${\tau }_3((n_1,n_2))$$

$$=(\zeta (2){)}^2\sum _{q_1,q_2=1}^{\infty }\frac{\tau ([q_1,q_2])({\varphi }_2([q_1,q_2]){)}^2}{[q_1,q_2{]}^6}{c}_{q_1}^{\ast }(n_1)c_{q_2}^{\ast }(n_2)(n_1,n_2\in \text{mathds}N).$$

$$(k,n{)}_{\ast }=\max \{d:d\mid k,d\mid \mid n\}.$$

$$(f\times g)(n)=\sum _{d\mid \mid n}f(d)g(n/d)(n\in \text{mathds}N),$$

$$c_q^{\ast }(n)=\sum _{\begin{array}{c}1\le k\le q\\ (k,q{)}_{\ast }=1\end{array}}\exp (2\pi ikn/q)(q,n\in \text{mathds}N).$$

$$c_q^{\ast }(n)=\sum _{d\mid \mid (n,q{)}_{\ast }}d{\mu }^{\ast }(q/d)(q,n\in \text{mathds}N),$$

$$\sum _{d\mid \mid q}{c}_d^{\ast }(n)=\{\begin{array}{ll}q,& \text{ if q∣n,}\\ 0,& \text{ otherwise}\end{array}$$

$$c_{p^{\nu }}^{\ast }(n)=\{\begin{array}{ll}{p}^{\nu }-1,& \text{ if pν∣n,}\\ -1,& \text{ otherwise,}\end{array}$$

$$\sum _{d\mid \mid q}|c_d^{\ast }(n)|=2^{\omega (q/(n,q{)}_{\ast })}(n,q{)}_{\ast },$$

$$\sum _{d\mid \mid q}|c_d^{\ast }(n)|\le 2^{\omega (q)}n.$$

$$\sum _{d\mid \mid p^{\nu }}|c_d^{\ast }(n)|=|c_1^{\ast }(n)|+|c_{p^{\nu }}^{\ast }(n)|=\{\begin{array}{ll}1+p^{\nu }-1=p^{\nu },& \text{ if pν∣n,}\\ 1+1=2,& \text{ otherwise.}\end{array}$$

$$\sum _{d\mid q}|c_d(n)|=2^{\omega (q/(n,q))}(n,q),$$

$$\sum _{d\mid q}|c_d(n)|\le 2^{\omega (q)}n,$$

$$\frac{1}{n}\sum _{k=1}^n{c}_{q_1}(k)c_{q_2}(k)=\{\begin{array}{ll}\varphi (q),& \text{ if q1=q2=q,}\\ 0,& \text{ otherwise,}\end{array}$$

$$\sum _{k=1}^{p^2}{c}_p^{\ast }(k)c_{p^2}^{\ast }(k)=p^2(p-1)\ne 0.$$

$${\Lambda }^{\ast }(k)=\{\begin{array}{ll}\nu \log p,& \text{ if k=pν,}\\ 0,& \text{ otherwise.}\end{array}$$

$$\sum _{n=1}^{\infty }\frac{c_q^{\ast }(n)}{n}=-{\Lambda }^{\ast }(q)(q\ge 2),$$

$$\sum _{n=1}^{\infty }\frac{c_q(n)}{n}=-\Lambda (q)(q\ge 2).$$

$$(f\ast g)(n_1,\dots ,n_k)=\sum _{d_1\mid n_1,\dots ,d_k\mid n_k}f(d_1,\dots ,d_k)g(n_1/d_1,\dots ,n_k/d_k),$$

Full text

The Ramanujan Journal 47 (2018), Issue 3, 589–603

2010 Mathematics Subject Classification: 11A25, 11N37

Key Words and Phrases: Ramanujan expansion of arithmetic functions, arithmetic function of several variables, multiplicative function, unitary divisor, unitary Ramanujan sum

1 Introduction

Basic notations, used throughout the paper are included in Section 2. Further notations are explained by their first appearance. Refining results of Wintner [25], Delange [3] proved the following general theorem concerning Ramanujan expansions of arithmetic functions.

Theorem 1**.**

Let $f:{\mathds{N}}\to{\mathds{C}}$ be an arithmetic function. Assume that

[TABLE]

Then for every $n\in{\mathds{N}}$ we have the absolutely convergent Ramanujan expansion

[TABLE]

where the coefficients $a_{q}$ are given by

[TABLE]

Delange also pointed out how this result can be formulated for multiplicative functions $f$. By Wintner’s theorem ([25, Part I], also see, e.g., Postnikov [11, Ch.  3], Schwarz and Spilker [12, Ch.  II]), condition (1) ensures that the mean value

[TABLE]

exists and $a_{1}=M(f)$.

Recently, Ushiroya [23] proved the analog of Theorem 1 for arithmetic functions of two variables and as applications, derived Ramanujan expansions of certain special functions. For example, he deduced that for any fixed $n_{1},n_{2}\in{\mathds{N}}$,

[TABLE]

[TABLE]

Here (3) corresponds to the classical identity

[TABLE]

due to Ramanujan [13]. However, (4) has not a direct one dimensional analog. The identity of Ramanujan for the divisor function, namely,

[TABLE]

cannot be obtained by the same approach, since the mean value $M(\tau)$ does not exist.

We remark that prior to Delange’s paper, Cohen [2] pointed out how absolutely convergent expansions (2), including (5) can be deduced for some special classes of multiplicative functions of one variable.

It is easy to see that the same arguments of the papers by Delange [3] and Ushiroya [23] lead to the extension of Theorem 1 for Ramanujan expansions of (multiplicative) arithmetic functions of $k$ variables, for any $k\in{\mathds{N}}$. But in order to obtain $k$ dimensional versions of the identities (3) and (4), the method given in [23] to compute the coefficients is complicated.

Recall that $d$ is a unitary divisor (or block divisor) of $n$ if $d\mid n$ and $(d,n/d)=1$. Various problems concerning functions associated to unitary divisors were discussed by several authors in the literature. Many properties of the functions $\sigma^{*}(n)$ and $\tau^{*}(n)=2^{\omega(n)}$, representing the sum, respectively the number of unitary divisors of $n$ are similar to $\sigma(n)$ and $\tau(n)$. Analogs of Ramanujan sums defined by unitary divisors, denoted by $c^{*}_{q}(n)$ and called unitary Ramanujan sums, are also known in the literature. However, we are not aware of any paper concerning expansions of functions with respect to the sums $c^{*}_{q}(n)$. The only such paper we found is by Subbarao [17], but it deals with a series over the other argument, namely $n$ in $c^{*}_{q}(n)$. See Section 3.2.

In this paper we show that certain properties on expansions of functions of one and several variables using classical and unitary Ramanujan sums, respectively, run parallel. We also present a simple approach to deduce Ramanujan expansions of both types for $g((n_{1},\ldots,n_{k}))$, where $g:{\mathds{N}}\to{\mathds{C}}$ is a certain function. For example, we have the identities

[TABLE]

[TABLE]

[TABLE]

which can be compared to (5), (3) and (4), respectively.

We point out that there are identities concerning the sums $c^{*}_{q}(n)$, which do not have simple counterparts in the classical case. For example,

[TABLE]

[TABLE]

For general accounts on classical Ramanujan sums and Ramanujan expansions of functions of one variable we refer to the book by Schwarz and Spilker [12] and to the survey papers by Lucht [8] and Ram Murty [14]. See Sections 3.1 and 3.2 for properties and references on functions defined by unitary divisors and unitary Ramanujan sums. Section 3.3 includes the needed background on arithmetic functions of several variables. Our main results and their proofs are presented in Section 4.

2 Notations

$\bullet$ ${\mathds{N}}=\{1,2,\ldots\}$, ${\mathds{N}}_{0}=\{0,1,2,\ldots\}$,

$\bullet$ ${\mathds{P}}$ is the set of (positive) primes,

$\bullet$ the prime power factorization of $n\in{\mathds{N}}$ is $n=\prod_{p\in{\mathds{P}}}p^{\nu_{p}(n)}$, the product being over the primes $p$, where all but a finite number of the exponents $\nu_{p}(n)$ are zero,

$\bullet$ $(n_{1},\ldots,n_{k})$ and $[n_{1},\ldots,n_{k}]$ denote the greatest common divisor and the least common multiple, respectively of $n_{1},\ldots,n_{k}\in{\mathds{N}}$,

$\bullet$ $(f*g)(n)=\sum_{d\mid n}f(d)g(n/d)$ is the Dirichlet convolution of the functions $f,g:{\mathds{N}}\to{\mathds{C}}$,

$\bullet$ $\omega(n)$ stands for the number of distinct prime divisors of $n$,

$\bullet$ $\mu$ is the Möbius function,

$\bullet$ $\sigma_{s}(n)=\sum_{d\mid n}d^{s}$ ($s\in{\mathbb{R}}$),

$\bullet$ $\sigma(n)=\sigma_{1}(n)$ is the sum of divisors of $n$,

$\bullet$ $\tau(n)=\sigma_{0}(n)$ is the number of divisors of $n$,

$\bullet$ $\phi_{s}$ is the Jordan function of order $s$ given by $\phi_{s}(n)=n^{s}\prod_{p\mid n}(1-1/p^{s})$ ($s\in{\mathbb{R}}$),

$\bullet$ $\phi=\phi_{1}$ is Euler’s totient function,

$\bullet$ $\psi_{s}$ is the generalized Dedekind function given by $\psi_{s}(n)=n^{s}\prod_{p\mid n}(1+1/p^{s})$ ($s\in{\mathbb{R}}$),

$\bullet$ $\psi=\psi_{1}$ is the Dedekind function,

$\bullet$ $c_{q}(n)=\sum_{1\leq k\leq q,(k,q)=1}\exp(2\pi ikn/q)$ are the Ramanujan sums ($q,n\in{\mathds{N}}$),

$\bullet$ $\tau_{k}(n)$ is the Piltz divisor function, defined as the number of ways that $n$ can be written as a product of $k$ positive integers,

$\bullet$ $\lambda(n)=(-1)^{\Omega(n)}$ is the Liouville function, where $\Omega(n)=\sum_{p}\nu_{p}(n)$,

$\bullet$ $\zeta$ is the Riemann zeta function,

$\bullet$ $d\mid\mid n$ means that $d$ is a unitary divisor of $n$, i.e., $d\mid n$ and $(d,n/d)=1$. (We remark that this is in concordance with the standard notation $p^{\nu}\mid\mid n$ used for prime powers $p^{\nu}$.)

3 Preliminaries

3.1 Functions of one variable defined by unitary divisors

The study of arithmetic functions defined by unitary divisors goes back to Vaidyanathaswamy [24] and Cohen [1]. The functions $\sigma^{*}(n)$ and $\tau^{*}(n)$ were already mentioned in the Introduction. The analog of Euler’s $\phi$ function is $\phi^{*}$, defined by $\phi^{*}(n)=\#\{k\in{\mathds{N}}:1\leq k\leq n,(k,n)_{*}=1\}$, where

[TABLE]

Note that $d\mid\mid(k,n)_{*}$ holds if and only if $d\mid k$ and $d\mid\mid n$. The functions $\sigma^{*}$, $\tau^{*}$, $\phi^{*}$ are all multiplicative and $\sigma^{*}(p^{\nu})=p^{\nu}+1$, $\tau^{*}(p^{\nu})=2$, $\phi^{*}(p^{\nu})=p^{\nu}-1$ for any prime powers $p^{\nu}$ ($\nu\geq 1$).

The unitary convolution of the functions $f$ and $g$ is

[TABLE]

it preserves the multiplicativity of functions, and the inverse of the constant $1$ function under the unitary convolution is $\mu^{*}$, where $\mu^{*}(n)=(-1)^{\omega(n)}$, also multiplicative. The set ${\cal A}$ of arithmetic functions forms a unital commutative ring with pointwise addition and the unitary convolution, having divisors of zero.

See, e.g., Derbal [4], McCarthy [10], Sitaramachandrarao and Suryanarayana [15], Snellman [16] for properties and generalizations of functions associated to unitary divisors.

3.2 Unitary Ramanujan sums

The unitary Ramanujan sums $c^{*}_{q}(n)$ were defined by Cohen [1] as follows:

[TABLE]

The identities

[TABLE]

[TABLE]

can be compared to the corresponding ones concerning the Ramanujan sums $c_{q}(n)$.

It turns out that $c^{*}_{q}(n)$ is multiplicative in $q$ for any fixed $n\in{\mathds{N}}$, and

[TABLE]

for any prime powers $p^{\nu}$ ($\nu\geq 1$). Furthermore, $c^{*}_{q}(q)=\phi^{*}(q)$, $c^{*}_{q}(1)=\mu^{*}(q)$ ($q\in{\mathds{N}}$).

If $q$ is squarefree, then the divisors of $q$ coincide with its unitary divisors. Therefore, $c^{*}_{q}(n)=c_{q}(n)$ for any squarefree $q$ and any $n\in{\mathds{N}}$. However, if $q$ is not squarefree, then there is no direct relation between these two sums.

Proposition 1**.**

For any $q,n\in{\mathds{N}}$,

[TABLE]

[TABLE]

Proof.

If $q=p^{\nu}$ ($\nu\geq 1$) is a prime power, then we have by (10),

[TABLE]

Now (11) follows at once by the multiplicativity in $q$ of the involved functions, while (12) is its immediate consequence. ∎

Remark 1**.**

The corresponding properties for the classical Ramanujan sums are

[TABLE]

[TABLE]

where inequality (14) is crucial in the proof of Theorem 1, and identity (13) was pointed out by Grytczuk [5]. **

Of course, some other properties differ notably. For example, the unitary sums $c^{*}_{q}(n)$ do not enjoy the orthogonality property of the classical Ramanujan sums, namely

[TABLE]

valid for any $q_{1},q_{2},n\in{\mathds{N}}$ with $[q_{1},q_{2}]\mid n$. To see this, let $p$ be a prime, let $q_{1}=p$, $q_{2}=p^{2}$ and $n=p^{2}$. Then, according to (10),

[TABLE]

Let $\Lambda^{*}$ be the unitary analog of the von Mangoldt function $\Lambda$, given by

[TABLE]

The paper of Subbarao [17] includes the formula

[TABLE]

with an incomplete proof, namely without showing that the series in (15) converges. Here (15) is the analog of Ramanujan’s formula (also see Hölder [6])

[TABLE]

For further properties and generalizations of unitary Ramanujan sums, see, e.g., Cohen [1], Johnson [7], McCarthy [10], Suryanarayana [18]. Note that a common generalization of the sums $c_{q}(n)$ and $c^{*}_{q}(n)$, involving Narkiewicz type regular systems of divisors was investigated by McCarthy [9] and the author [19].

3.3 Arithmetic functions of several variables

For every fixed $k\in{\mathds{N}}$ the set ${\cal A}_{k}$ of arithmetic functions $f:{\mathds{N}}^{k}\to{\mathds{C}}$ of $k$ variables is an integral domain with pointwise addition and the Dirichlet convolution defined by

[TABLE]

the unity being the function $\delta_{k}$, where

[TABLE]

The inverse of the constant $1$ function under (16) is $\mu_{k}$, given by

[TABLE]

where $\mu$ is the (classical) Möbius function.

A function $f\in{\cal A}_{k}$ is said to be multiplicative if it is not identically zero and

[TABLE]

holds for any $m_{1},\ldots,m_{k},n_{1},\ldots,n_{k}\in{\mathds{N}}$ such that $(m_{1}\cdots m_{k},n_{1}\cdots n_{k})=1$.

If $f$ is multiplicative, then it is determined by the values $f(p^{\nu_{1}},\ldots,p^{\nu_{k}})$, where $p$ is prime and $\nu_{1},\ldots,\nu_{k}\in{\mathds{N}}_{0}$. More exactly, $f(1,\ldots,1)=1$ and for any $n_{1},\ldots,n_{k}\in{\mathds{N}}$,

[TABLE]

Similar to the one dimensional case, the Dirichlet convolution (16) preserves the multiplicativity of functions. The unitary convolution for the $k$ variables case can be defined and investigated, as well, but we will not need it in what follows. See our paper [21], which is a survey on (multiplicative) arithmetic functions of several variables.

4 Main results

We first prove the following general result:

Theorem 2**.**

Let $f:{\mathds{N}}^{k}\to{\mathds{C}}$ be an arithmetic function ($k\in{\mathds{N}}$). Assume that

[TABLE]

Then for every $n_{1},\ldots,n_{k}\in{\mathds{N}}$,

[TABLE]

and

[TABLE]

where

[TABLE]

[TABLE]

the series (18) and (19) being absolutely convergent.

We remark that according to the generalized Wintner theorem, under conditions of Theorem 2, the mean value

[TABLE]

exists and $a_{1,\ldots,1}=a^{*}_{1,\ldots,1}=M(f)$. See Ushiroya [22, Th. 1] and the author [21, Sect. 7.1].

Proof.

We consider the case of the unitary Ramanujan sums $c^{*}_{q}(n)$. For any $n_{1},\ldots,n_{k}\in{\mathds{N}}$ we have, by using property (9),

[TABLE]

[TABLE]

[TABLE]

giving expansion (19) with the coefficients (20), by denoting $d_{1}=m_{1}q_{1},\ldots,d_{k}=m_{k}q_{k}$. The rearranging of the terms is justified by the absolute convergence of the multiple series, shown hereinafter:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

by using inequality (12) and condition (17).

For the Ramanujan sums $c_{q}(n)$ the proof is along the same lines, by using inequality (14). In the case $k=2$ the proof of (18) was given by Ushiroya [23]. ∎

For multiplicative functions $f$, condition (17) is equivalent to

[TABLE]

and to

[TABLE]

We deduce the following result:

Corollary 1**.**

Let $f:{\mathds{N}}^{k}\to{\mathds{C}}$ be a multiplicative function ($k\in{\mathds{N}}$). Assume that condition (21) or (22) holds. Then for every $n_{1},\ldots,n_{k}\in{\mathds{N}}$ one has the absolutely convergent expansions (18), (19), and the coefficients can be written as

[TABLE]

[TABLE]

where $\sum^{{}^{\prime}}$ means that for fixed $p$ and $j$, $\nu_{j}$ takes all values $\nu_{j}\geq 0$ if $\nu_{p}(q_{j})=0$, and takes only the value $\nu_{j}=0$ if $\nu_{p}(q_{j})\geq 1$.

Proof.

This is a direct consequence of Theorem 2 and the definition of multiplicative functions. In the cases $k=1$ and $k=2$, with the sums $c_{q}(n)$ see [3, Eq. (7)] and [23, Th. 2.4], respectively. ∎

Next we consider the case $f(n_{1},\ldots,n_{k})=g((n_{1},\ldots,n_{k}))$.

Theorem 3**.**

Let $g:{\mathds{N}}\to{\mathds{C}}$ be an arithmetic function and let $k\in{\mathds{N}}$. Assume that

[TABLE]

Then for every $n_{1},\ldots,n_{k}\in{\mathds{N}}$,

[TABLE]

[TABLE]

are absolutely convergent, where

[TABLE]

[TABLE]

with the notation $Q=[q_{1},\ldots,q_{k}]$.

Proof.

We apply Theorem 2. The identity

[TABLE]

shows that now

[TABLE]

In the unitary case the coefficients are

[TABLE]

[TABLE]

and take into account that $q_{1}\mid\mid n,\ldots,q_{k}\mid\mid n$ holds if and only if $[q_{1},\ldots,q_{k}]=Q\mid\mid n$, that is, $n=mQ$ with $(m,Q)=1$.

In the classical case, namely for the coefficients $a_{q_{1},\ldots,q_{k}}$, the proof is analog. ∎

If the function $g$ is multiplicative, then condition (23) is equivalent to

[TABLE]

and to

[TABLE]

and Theorem 3 can be rewritten according to Corollary 1. We continue, instead, with the following result, having direct applications to special functions.

Corollary 2**.**

Let $g:{\mathds{N}}\to{\mathds{C}}$ be a multiplicative function and let $k\in{\mathds{N}}$. Assume that condition (26) or (27) holds.

If $\mu*g$ is completely multiplicative, then (24) holds for every $n_{1},\ldots,n_{k}\in{\mathds{N}}$ with

[TABLE]

Identity (25) holds for every $n_{1},\ldots,n_{k}\in{\mathds{N}}$ (and any multiplicative $g$) with

[TABLE]

Proof.

This is immediate by Theorem 3. By making use of condition $(m,Q)=1$, it follows for the coefficients $a^{*}_{q_{1},\ldots,q_{k}}$ that $(\mu*g)(mQ)=(\mu*g)(m)(\mu*g)(Q)$ for any multiplicative function $g$. ∎

Corollary 3**.**

For every $n_{1},\ldots,n_{k}\in{\mathds{N}}$ the following series are absolutely convergent:

[TABLE]

[TABLE]

[TABLE]

Proof.

Apply Corollary 2 for $g(n)=\sigma_{s}(n)/n^{s}$, where the function $(\mu*g)(n)=1/n^{s}$ is completely multiplicative. ∎

In the case $k=2$ the identities of Corollary 3 were given in [23, Ex. 3.8, 3.9]. For $k=2$, (28) and (29) reduce to (3) and (4), respectively.

Corollary 4**.**

For every $n_{1},\ldots,n_{k}\in{\mathds{N}}$ the following series are absolutely convergent:

[TABLE]

with $s\in{\mathbb{R}},s+k>1$,

[TABLE]

[TABLE]

Proof.

Apply the second part of Corollary 2 for $g(n)=\sigma_{s}(n)/n^{s}$. Since $(\mu*g)(n)=1/n^{s}$, we deduce that

[TABLE]

∎

If $k=2$, then identities (30) and (31) recover (6) and (7), respectively.

Corollary 2 can be applied for several other special functions $g$. Further examples are $g(n)=\beta_{s}(n)/n^{s}$ ($s\in{\mathbb{R}}$) and $g(n)=r(n)$, where $\beta_{s}(n)=\sum_{d\mid n}d^{s}\lambda(n/d)$ and $r(n)$ denotes the number of solutions $(x,y)\in{\mathds{Z}}^{2}$ of the equation $x^{2}+y^{2}=n$.

Corollary 5**.**

For every $n_{1},\ldots,n_{k}\in{\mathds{N}}$ and $s\in{\mathbb{R}},s+k>1$ the following series are absolutely convergent:

[TABLE]

[TABLE]

In the case $k=s=1$, identity (32) was derived by Cohen [2, Eq. (12)]. See also the author [20, Eq. (16)].

Corollary 6**.**

For every $n_{1},\ldots,n_{k}\in{\mathds{N}}$ ($k\geq 1$) the following series are absolutely convergent:

[TABLE]

[TABLE]

where

[TABLE]

$\chi=\chi_{4}$* being the nonprincipal character (mod $4$) and*

[TABLE]

Proof.

Use that $r(n)=4\sum_{d\mid n}\chi(d)$, thus $\mu*r/4=\chi$, completely multiplicative. Here (33) is well known for $k=1$, and was obtained in [23, Ex. 3.13] in the case $k=2$. ∎

If $g$ is multiplicative such that $\mu*g$ is not completely multiplicative, then Corollary 2 can be applied for the sums $c^{*}_{q}(n)$, but not for $c_{q}(n)$, in general.

Corollary 7**.**

For every $n_{1},\ldots,n_{k}\in{\mathds{N}}$ the following series are absolutely convergent:

[TABLE]

with $s\in{\mathbb{R}},s+k>1$,

[TABLE]

Proof.

Similar to the proof of Corollary 4, selecting $g(n)=\phi_{s}(n)/n^{s}$, where $(\mu*g)(n)=\mu(n)/n^{s}$. ∎

However, observe that if $Q$ is not squarefree, then $\mu(Q)=0$ and all terms of the sums in (34) and (35) are zero. Now if $Q$ is squarefree, then $q_{j}$ is squarefree and $c^{*}_{q_{j}}(n_{j})=c_{q_{j}}(n_{j})$ for any $j$ and any $n_{j}$. Thus we deduce the next identities:

Corollary 8**.**

For every $n_{1},\ldots,n_{k}\in{\mathds{N}}$ the following series are absolutely convergent:

[TABLE]

with $s\in{\mathbb{R}},s+k>1$,

[TABLE]

which are formally the same identities as (34) and (35).

Note that the special identities (36) and (37) can also be derived by the first part of Corollary 2, without considering the unitary sums $c^{*}_{q}(n)$. See Ushiroya [23, Ex. 3.10, 3.11] for the case $k=2$, using different arguments.

Similar formulas can be deduced for the generalized Dedekind function $\psi_{s}(n)=n^{s}\prod_{p\mid n}(1+1/p^{s})$ ($s\in{\mathbb{R}}$). See Cohen [2, Eq. (13)] in the case $k=s=1$.

Finally, consider the Piltz divisor function $\tau_{m}$. The following formula, concerning the sums $c^{*}_{q}(n)$, deduced along the same lines with the previous ones, has no simple counterpart in the classical case.

Corollary 9**.**

For any $n_{1},\ldots,n_{k},m\in{\mathds{N}}$ with $m\geq 2,k\geq 2$,

[TABLE]

If $m=2$, then (38) reduces to formula (31), while for $m=3$ and $k=2$ it gives (8).

5 Acknowledgement

The author thanks the anonymous referee for careful reading of the manuscript and helpful comments.