Short Proofs, Generalizations, and Applications of Certain Identities Concerning Multiple Dirichlet Series

László Tóth Department of Mathematics University of Pécs Ifjúság útja 6 7624 Pécs Hungary

                    ltoth@gamma.ttk.pte.hu

Abstract We present other proofs, generalizations, and analogues of the identities concerning multiple Dirichlet series by Tahmi and Derbal (2022). As applications, we obtain asymptotic formulas with remainder terms for certain related sums.

1 Introduction

Recently, Tahmi and Derbal [4] obtained certain identities for the multiple Dirichlet series

∑_{\begin{matrix}n_{1},...,n_{r}=1 \\ gcd(n_{1},...,n_{r})=1 \end{matrix}}^{∞} \frac{f(n_{1}) ⋯ f(n_{r})}{n_{1}^{s_{1}} ⋯ n_{r}^{s_{r}}},

with r ≥ 2 an integer, in the cases where f : ℕ := {1, 2, …} → ℂ is a completely multiplicative arithmetic function or the Dirichlet convolution of two completely multiplicative functions.

As direct corollaries of their results, they mentioned, among others, the identities

∑_{\begin{matrix}n_{1},...,n_{r}=1 \\ gcd(n_{1},...,n_{r})=1 \end{matrix}}^{∞} \frac{1}{n_{1}^{s_{1}} ⋯ n_{r}^{s_{r}}} = \frac{ζ(s_{1}) ⋯ ζ(s_{r})}{ζ(s_{1} + ⋯ + s_{r})},

with s_{i} ∈ ℂ, ℜs_{i} > 1 (1 ≤ i ≤ r), concerning the Dirichlet series of the characteristic function of the set of points in ℕ^{r}, which are visible from the origin (also see Apostol [1, p. 248]), and

∑_{\begin{matrix}n_{1},n_{2}=1 \\ gcd(n_{1},n_{2})=1 \end{matrix}}^{∞} \frac{τ(n_{1})τ(n_{2})}{n_{1}^{s_{1}}n_{2}^{s_{2}}} = ζ^{2}(s_{1})ζ^{2}(s_{2}) ∏_{p}(1 − \frac{4}{p^{s_{1}+s_{2}}} + \frac{2}{p^{2s_{1}+s_{2}}} + \frac{2}{p^{s_{1}+2s_{2}}} − \frac{1}{p^{2s_{1}+2s_{2}}} ), (1)

with s_{i} ∈ ℂ, ℜs_{i} > 1 (1 ≤ i ≤ 2), where τ(n) = ∑_{d|n} 1 is the divisor function.

The proofs given in [4] are by using Euler product expansions of the Dirichlet series of some appropriate multiplicative functions of one variable.

In this paper we present other proofs and generalizations of the results by Tahmi and Derbal [4] by considering Euler product expansions of some multiple Dirichlet series

∑_{n_{1},...,n_{r}=1}^{∞} \frac{F (n_{1}, …, n_{r})}{n_{1}^{s_{1}} ⋯ n_{r}^{s_{r}}}

of multiplicative arithmetic functions F : ℕ^{r} → ℂ of r variables. Namely, we investigate the functions

F (n_{1}, …, n_{r}) = \begin{cases}f_{1}(n_{1}) ⋯ f_{r}(n_{r}), & if gcd(n_{1}, …, n_{r}) = 1; \\ 0, & otherwise, \end{cases}  (2)

where each of the functions f_{1}, …, f_{r} : ℕ → ℂ is the Dirichlet convolution of t (t ≥ 1) completely multiplicative functions. Note that if f_{1}, …, f_{r} are multiplicative, then F given by (2) is multiplicative, viewed as a function of r variables.

We also make more explicit the formula of [4, Th. 3.2], as applications we obtain asymptotic formulas with remainder terms for certain sums

∑_{\begin{matrix}n_{1},...,n_{r}≤x \\ gcd(n_{1},...,n_{r})=1 \end{matrix}} f_{1}(n_{1}) ⋯ f_{r}(n_{r}),

and also derive similar results where the condition gcd(n_{1}, …, n_{r}) = 1 is replaced by the condition that n_{1}, …, n_{r} are pairwise relatively prime. Some basic properties of multiplicative arithmetic functions of r variables are reviewed in Section 2.1. Certain polynomial identities needed in the proofs are given in Section 2.2. Our main results and their proofs on the Dirichlet series are included in Section 3, and some related asymptotic formulas are presented in Section 4. All the identities regarding Dirichlet series and Euler products are considered formally or in the case of absolute convergence.

2 Preliminaries

2.1 Arithmetic functions of several variables

Let F : ℕ^{r} → ℂ be an arbitrary arithmetic function of r variables (r ≥ 1). Its Dirichlet series is given by

D(F, s_{1}, …, s_{r}) := ∑_{n_{1},...,n_{r}=1}^{∞} \frac{F (n_{1}, …, n_{r})}{n_{1}^{s_{1}} ⋯ n_{r}^{s_{r}}}.

The Dirichlet convolution of the functions F, G : ℕ^{r} → ℂ is defined by

(F ∗ G)(n_{1}, …, n_{r}) = ∑_{d_{1}|n_{1},...,d_{r}|n_{r}} F (d_{1}, …, d_{r})G(n_{1}/d_{1}, …, n_{r}/d_{r}).

If D(F, s_{1}, …, s_{r}) and D(G, s_{1}, …, s_{r}), with s_{1}, …, s_{r} ∈ ℂ, are absolutely convergent, then D(F ∗ G; s_{1}, …, s_{r}) is also absolutely convergent and

D(F ∗ G, s_{1}, …, s_{r}) = D(F, s_{1}, …, s_{r})D(G, s_{1}, …, s_{r}).

A nonzero arithmetic function F : ℕ^{r} → ℂ is said to be multiplicative if

F (m_{1}n_{1}, …, m_{r}n_{r}) = F (m_{1}, …, m_{r})F (n_{1}, …, n_{r})

holds for every m_{1}, …, m_{r}, n_{1}, …, n_{r} ∈ ℕ such that gcd(m_{1} ⋯ m_{r}, n_{1} ⋯ n_{r}) = 1. If F is multiplicative, then it is determined by the values F (p^{a_{1}}, …, p^{a_{r}}), where p is prime and a_{1}, …, a_{r} ∈ ℕ_{0} := ℕ ∪ {0}. More exactly, F (1, …, 1) = 1 and for every n_{1}, …, n_{r} ∈ ℕ,

F (n_{1}, …, n_{r}) = ∏_{p} F (p^{a_{p}(n_{1})}, …, p^{a_{p}(n_{r})}),

by using the notation n = ∏_{p} p^{a_{p}(n)} for the prime power factorization of n ∈ ℕ, the product being over the primes p, where all but a finite number of the exponents a_{p}(n) are zero.

Examples of multiplicative functions of r variables are the GCD and LCM functions gcd(n_{1}, …, n_{r}), lcm(n_{1}, …, n_{r}) and the characteristic functions

ϱ(n_{1}, …, n_{r}) = \begin{cases}1, & if gcd(n_{1}, …, n_{r}) = 1; \\ 0, & otherwise, \end{cases}

ϑ(n_{1}, …, n_{r}) = \begin{cases}1, & if gcd(n_{i}, n_{j}) = 1 for every 1 ≤ i < j ≤ r; \\ 0, & otherwise. \end{cases}

If F, G : ℕ^{r} → ℂ are multiplicative, then their Dirichlet convolution F ∗ G is also multiplicative. If F is multiplicative, then its Dirichlet series can be expanded into a (formal) Euler product, that is,

D(F, s_{1}, …, s_{r}) = ∏_{p} ∑_{a_{1},...,a_{r}=0}^{∞} \frac{f(p^{a_{1}}, …, p^{a_{r}})}{p^{a_{1}s_{1}+···+a_{r}s_{r}}},  (3)

the product being over the primes p. More exactly, if F is multiplicative, then the series D(F, s_{1}, …, s_{r}) with s_{1}, …, s_{r} ∈ ℂ is absolutely convergent if and only if

∑_{p} ∑_{\begin{matrix}a_{1},...,a_{r}=0 \\ a_{1}+···+a_{r}≥1 \end{matrix}}^{∞} \frac{|f(p^{a_{1}}, …, p^{a_{r}})|}{p^{a_{1}ℜs_{1}+···+a_{r}ℜs_{r}}} < ∞

and in this case equality (3) holds.

See, e.g., Delange [2] and the survey by the author [5] for these and some related results on arithmetic functions of r variables. If r = 1, i.e., in the case of functions of a single variable we recover some familiar properties.

2.2 Some polynomial identities

Let e_{j}(x_{1}, …, x_{t}) = ∑_{1≤i_{1}<···<i_{j}≤t} x_{i_{1}} ⋯ x_{i_{j}} denote the elementary symmetric polynomials in x_{1}, …, x_{t} of degree j (1 ≤ j ≤ t). We will use the polynomial identity

P (x) := ∏_{j=1}^{t}(x − x_{j}) = x^{t} + ∑_{j=1}^{t}(−1)^{j}e_{j}(x_{1}, …, x_{t})x^{t−j}.  (4)

Taking derivatives gives

P′(x) = ∑_{j=1}^{t} ∏_{\begin{matrix}k=1 \\ k≠j \end{matrix}}^{t}(x − x_{k}) = tx^{t−1} + ∑_{j=1}^{t−1}(−1)^{j}(t − j)e_{j}(x_{1}, …, x_{t})x^{t−j−1}.  (5)

We need the following lemma. Lemma 1. If t ∈ ℕ and x_{1}, …, x_{t} ∈ ℂ, then

(1 − t) ∏_{j=1}^{t}(1 − x_{j}) + ∑_{j=1}^{t} ∏_{\begin{matrix}k=1 \\ k≠j \end{matrix}}^{t}(1 − x_{k}) = 1 + ∑_{j=2}^{t}(−1)^{j−1}(j − 1)e_{j}(x_{1}, …, x_{t}).

Proof. By using (4) and (5),

(1 − t) ∏_{j=1}^{t}(1 − x_{j}) + ∑_{j=1}^{t} ∏_{\begin{matrix}k=1 \\ k≠j \end{matrix}}^{t}(1 − x_{k}) = (1 − t)P (1) + P′(1)

= (1 − t)(1 + ∑_{j=1}^{t}(−1)^{j}e_{j}(x_{1}, …, x_{t}))

+ t + ∑_{j=1}^{t−1}(−1)^{j}(t − j)e_{j}(x_{1}, …, x_{t})

= 1 + ∑_{j=2}^{t}(−1)^{j−1}(j − 1)e_{j}(x_{1}, …, x_{t}).

3 Identities for Dirichlet series

Our first result is the following. As above, let ∗ denote the Dirichlet convolution of arithmetic functions and let D(f, s) := ∑_{n=1}^{∞} f(n)n^{−s} stand for the Dirichlet series of the function f : ℕ → ℂ. We recall that a nonzero function f : ℕ → ℂ is completely multiplicative if f(mn) = f(m)f(n) holds for all m, n ∈ ℕ.

Theorem 2. Let r ≥ 2, t ≥ 1 be fixed integers and let f_{i} = g_{i1} ∗ ⋯ ∗ g_{it}, where g_{ij} : ℕ → ℂ are nonzero completely multiplicative functions (1 ≤ i ≤ r, 1 ≤ j ≤ t). Then we have (formally or in the case of absolute convergence),

∑_{\begin{matrix}n_{1},...,n_{r}=1 \\ gcd(n_{1},...,n_{r})=1 \end{matrix}}^{∞} \frac{f_{1}(n_{1}) ⋯ f_{r}(n_{r})}{n_{1}^{s_{1}} ⋯ n_{r}^{s_{r}}} = D(f_{1}, s_{1}) ⋯ D(f_{r}, s_{r})Δ(f_{1}, …, f_{r}, s_{1}, …, s_{r}),

where

Δ(f_{1}, …, f_{r}, s_{1}, …, s_{r}) =
∏_{p}(1 + (−1)^{r−1} ∑_{1≤a_{1},...,a_{r}≤t} \frac{1}{p^{a_{1}s_{1}+···+a_{r}s_{r}}} ∏_{i=1}^{r}(−1)^{a_{i}}G_{ia_{i}}(p)),  (6)

and

G_{ij}(p) := ∑_{1≤ℓ_{1}<···<ℓ_{j}≤t} g_{iℓ_{1}}(p) ⋯ g_{iℓ_{j}}(p),  (7)

with 1 ≤ i ≤ r, 1 ≤ j ≤ t, and p a prime.

In the cases t = 1 and t = 2, with f_{1} = ⋯ = f_{r} = f, Theorem 2 recovers [4, Ths. 3.1, 3.2].

Proof. As mentioned above, the characteristic function ϱ of the r-tuples with relatively prime components is multiplicative, viewed as a functions of r variables. Note that for primes p and a_{1}, …, a_{r} ≥ 0 we have ϱ(p^{a_{1}}, …, p^{a_{r}}) = 1 if and only if there is at least one a_{i} = 0. Also, if f_{1}, …, f_{r} : ℕ → ℂ are arbitrary multiplicative functions of a single variable, then their product f_{1}(n_{1}) ⋯ f_{r}(n_{r}) is multiplicative as a function of r variables. We deduce the Euler product expansion

D := ∑_{\begin{matrix}n_{1},...,n_{r}=1 \\ gcd(n_{1},...,n_{r})=1 \end{matrix}}^{∞} \frac{f_{1}(n_{1}) ⋯ f_{r}(n_{r})}{n_{1}^{s_{1}} ⋯ n_{r}^{s_{r}}}

= ∑_{n_{1},...,n_{r}=1}^{∞} \frac{f_{1}(n_{1}) ⋯ f_{r}(n_{r})ϱ(n_{1}, …, n_{r})}{n_{1}^{s_{1}} ⋯ n_{r}^{s_{r}}}

= ∏_{p} ∑_{a_{1},...,a_{r}=0}^{∞} \frac{f_{1}(p^{a_{1}}) ⋯ f_{r}(p^{a_{r}})ϱ(p^{a_{1}}, …, p^{a_{r}})}{p^{a_{1}s_{1}+···+a_{r}s_{r}}}

= ∏_{p} ∑_{\begin{matrix}a_{1},...,a_{r}=0 \\ a_{1}···a_{r}=0 \end{matrix}}^{∞} \frac{f_{1}(p^{a_{1}}) ⋯ f_{r}(p^{a_{r}})}{p^{a_{1}s_{1}+···+a_{r}s_{r}}}

= ∏_{p}( ∑_{a_{1},...,a_{r}=0}^{∞} − ∑_{a_{1},...,a_{r}=1}^{∞})\frac{f_{1}(p^{a_{1}}) ⋯ f_{r}(p^{a_{r}})}{p^{a_{1}s_{1}+···+a_{r}s_{r}}}.

Now if f_{i} = g_{i1} ∗ ⋯ ∗ g_{it}, where g_{ij} : ℕ → ℂ are completely multiplicative functions (1 ≤ i ≤ r, 1 ≤ j ≤ t), then

∑_{n=1}^{∞} \frac{f_{i}(n)}{n^{s}} = ∏_{j=1}^{t} ∑_{n=1}^{∞} \frac{g_{ij}(n)}{n^{s}} = ∏_{j=1}^{t} ∏_{p}(1 − \frac{g_{ij}(p)}{p^{s}})^{−1} = ∏_{p} ∏_{j=1}^{t}(1 − \frac{g_{ij}(p)}{p^{s}})^{−1}.

At the same time, since the functions f_{i} (1 ≤ i ≤ r) are multiplicative, we have

∑_{n=1}^{∞} \frac{f_{i}(n)}{n^{s}} = ∏_{p} ∑_{a=0}^{∞} \frac{f_{i}(p^{a})}{p^{as}},

showing that

∑_{a=0}^{∞} \frac{f_{i}(p^{a})}{p^{as}} = ∏_{j=1}^{t}(1 − \frac{g_{ij}(p)}{p^{s}})^{−1},

and

∑_{a=1}^{∞} \frac{f_{i}(p^{a})}{p^{as}} = ∏_{j=1}^{t}(1 − \frac{g_{ij}(p)}{p^{s}})^{−1} − 1.  (8)

It follows that

D = ∏_{p}(∏_{i=1}^{r} ∏_{j=1}^{t}(1 − \frac{g_{ij}(p)}{p^{s_{i}}})^{−1} − ∏_{i=1}^{r}(∏_{j=1}^{t}(1 − \frac{g_{ij}(p)}{p^{s_{i}}})^{−1} − 1))

= ∏_{p} ∏_{i=1}^{r} ∏_{j=1}^{t}(1 − \frac{g_{ij}(p)}{p^{s_{i}}})^{−1} ∏_{p}(1 − ∏_{i=1}^{r}(1 − ∏_{j=1}^{t}(1 − \frac{g_{ij}(p)}{p^{s_{i}}}))).  (9)

Using identity (4) for x = 1 and x_{j} = g_{ij}(p)p^{−s_{i}} (1 ≤ j ≤ t) we have

1 − ∏_{j=1}^{t}(1 − \frac{g_{ij}(p)}{p^{s_{i}}}) = ∑_{j=1}^{t} \frac{(−1)^{j−1}}{p^{js_{i}}} ∑_{1≤ℓ_{1}<···<ℓ_{j}≤t} g_{iℓ_{1}}(p) ⋯ g_{iℓ_{j}}(p),

and inserting into (9) we deduce

D = D(f_{1}, s_{1}) ⋯ D(f_{r}, s_{r}) ∏_{p}(1 − ∏_{i=1}^{r} ∑_{j=1}^{t} \frac{(−1)^{j−1}}{p^{js_{i}}} G_{ij}(p)),

where G_{ij}(p) is defined by (7). Here the product over the primes p is

∏_{p}(1 − (∑_{a_{1}=1}^{t} \frac{(−1)^{a_{1}−1}}{p^{a_{1}s_{1}}} G_{1a_{1}}(p)) ⋯ (∑_{a_{r}=1}^{t} \frac{(−1)^{a_{r}−1}}{p^{a_{r}s_{r}}} G_{ra_{r}}(p)))

= ∏_{p}(1 + (−1)^{r−1} ∑_{1≤a_{1},...,a_{r}≤t} \frac{1}{p^{a_{1}s_{1}+···+a_{r}s_{r}}} ∏_{i=1}^{r}(−1)^{a_{i}}G_{ia_{i}}(p)),

finishing the proof.

Remark 3. Identity (6) shows that under the assumptions of Theorem 2 we have the convolutional identity

f_{1}(n_{1}) ⋯ f_{r}(n_{r})ϱ(n_{1}, …, n_{r}) = ∑_{d_{1}e_{1}=n_{1},...,d_{r}e_{r}=n_{r}} f_{1}(d_{1}) ⋯ f_{r}(d_{r})F_{f_{1},...,f_{r}}(e_{1}, …, e_{r}), (10)

where F_{f_{1},...,f_{r}} is the multiplicative function defined for prime powers p^{a_{1}}, …, p^{a_{r}} (a_{1}, …, a_{r} ≥ 0, not all zero) by

F_{f_{1},...,f_{r}}(p^{a_{1}}, …, p^{a_{r}}) = \begin{cases}(−1)^{r−1} ∏_{i=1}^{r}(−1)^{a_{i}}G_{ia_{i}}(p), & if 1 ≤ a_{1}, …, a_{r} ≤ t; \\ 0, & otherwise. \end{cases}

Let τ_{t}(n) = ∑_{d_{1}···d_{t}=n} 1 denote the Piltz divisor function of order t.

Corollary 4. Let r ≥ 2 and let t_{i} ≥ 2 (1 ≤ i ≤ r) be fixed integers. If s_{i} ∈ ℂ, ℜs_{i} > 1 (1 ≤ i ≤ r), then

∑_{\begin{matrix}n_{1},...,n_{r}=1 \\ gcd(n_{1},...,s_{r})=1 \end{matrix}}^{∞} \frac{τ_{t_{1}}(n_{1}) ⋯ τ_{t_{r}}(n_{r})}{n_{1}^{s_{1}} ⋯ n_{r}^{s_{r}}} = ζ^{t_{1}}(s_{1}) ⋯ ζ^{t_{r}}(s_{r})Δ(τ_{t_{1}}, …, τ_{t_{r}}, s_{1}, …, s_{r}),

with

Δ(τ_{t_{1}}, …, τ_{t_{r}}, s_{1}, …, s_{r}) = ∏_{p}(1 + (−1)^{r−1} ∑_{1≤a_{1},...,a_{r}≤t} \frac{1}{p^{a_{1}s_{1}+···+a_{r}s_{r}}} ∏_{i=1}^{r}(−1)^{a_{i}} \binom{t_{i}}{a_{i}}), (11)

where \binom{t_{i}}{a_{i}} are binomial coefficients with the usual convention that \binom{t_{i}}{a_{i}} = 0 for a_{i} > t_{i} (1 ≤ i ≤ r).

Proof. Choose g_{ij}(p) = 1 for 1 ≤ j ≤ t_{i} and g_{ij}(p) = 0 for t_{i} + 1 ≤ j ≤ t (1 ≤ i ≤ r). Then f_{i}(n) = τ_{t_{i}}(n) (1 ≤ i ≤ r), and use that G_{ij}(p) = \binom{t_{i}}{j} (1 ≤ i ≤ r, 1 ≤ j ≤ t) for a prime p.

Corollary 5. Let r ≥ 2. If s_{i} ∈ ℂ, ℜs_{i} > 1 (1 ≤ i ≤ r), then

∑_{\begin{matrix}n_{1},...,n_{r}=1 \\ gcd(n_{1},...,n_{r})=1 \end{matrix}}^{∞} \frac{τ(n_{1}) ⋯ τ(n_{r})}{n_{1}^{s_{1}} ⋯ n_{r}^{s_{r}}} =

ζ^{2}(s_{1}) ⋯ ζ^{2}(s_{r}) × ∏_{p}(1 + (−1)^{r−1} ∑_{1≤a_{1},...,a_{r}≤2} \frac{(−2)^{#\{1≤i≤r: a_{i}=1\}}}{p^{a_{1}s_{1}+···+a_{r}s_{r}}} ).

Proof. Apply Corollary 4 for t_{i} = 2 (1 ≤ i ≤ r).

If r = 2, then this recovers identity (1) and for r = 3 we have

∑_{\begin{matrix}n_{1},n_{2},n_{3}=1 \\ gcd(n_{1},n_{2},n_{3})=1 \end{matrix}}^{∞} \frac{τ(n_{1})τ(n_{2})τ(n_{3})}{n_{1}^{s_{1}}n_{2}^{s_{2}}n_{3}^{s_{3}}} = ζ^{2}(s_{1})ζ^{2}(s_{2})ζ^{2}(s_{3})

× ∏_{p}(1 − \frac{8}{p^{s_{1}+s_{2}+s_{3}}} + \frac{4}{p^{2s_{1}+s_{2}+s_{3}}} + \frac{4}{p^{s_{1}+2s_{2}+s_{3}}} + \frac{4}{p^{s_{1}+s_{2}+2s_{3}}}

−\frac{2}{p^{2s_{1}+2s_{2}+s_{3}}} − \frac{2}{p^{2s_{1}+s_{2}+2s_{3}}} − \frac{2}{p^{s_{1}+2s_{2}+2s_{3}}} + \frac{1}{p^{2s_{1}+2s_{2}+2s_{3}}} ).  (12)

Now we consider Dirichlet series with gcd(n_{1}, …, n_{r}) = 1 replaced by the condition that n_{1}, …, n_{r} are pairwise relatively prime.

Theorem 6. Let r ≥ 2, t ≥ 1 be fixed integers and let f_{i} = g_{i1} ∗ ⋯ ∗ g_{it}, where g_{ij} : ℕ → ℂ are nonzero completely multiplicative functions (1 ≤ i ≤ r, 1 ≤ j ≤ t). Then we have (formally or in the case of absolute convergence),

∑_{\begin{matrix}n_{1},...,n_{r}=1 \\ gcd(n_{i},n_{j})=1 (i≠j) \end{matrix}}^{∞} \frac{f_{1}(n_{1}) ⋯ f_{r}(n_{r})}{n_{1}^{s_{1}} ⋯ n_{r}^{s_{r}}} = D(f_{1}, s_{1}) ⋯ D(f_{r}, s_{r})\overline{Δ}(f_{1}, …, f_{r}, s_{1}, …, s_{r}),

where

\overline{Δ}(f_{1}, …, f_{r}, s_{1}, …, s_{r}) =

∏_{p}(1 − ∑_{i=2}^{r}(i − 1) ∑_{1≤ℓ_{1}<···<ℓ_{i}≤r} ∑_{a_{ℓ_{1}},...,a_{ℓ_{i}}=1}^{t} \frac{1}{p^{a_{ℓ_{1}}s_{ℓ_{1}}+···+a_{ℓ_{i}}s_{ℓ_{i}}}} ∏_{m=1}^{i}(−1)^{a_{ℓm}}G_{ia_{ℓm}}(p)), (13)

with G_{ij}(p) (1 ≤ i ≤ r, 1 ≤ j ≤ t, p prime) defined by (7).

Proof. The characteristic function ϑ of the r-tuples with pairwise relatively prime components is multiplicative, viewed as a functions of r variables. Note that for primes p and a_{1}, …, a_{r} ≥ 0 we have ϑ(p^{a_{1}}, …, p^{a_{r}}) = 1 if and only if there is at most one a_{i} ≥ 1. If f_{1}, …, f_{r} : ℕ → ℂ are arbitrary multiplicative functions of a single variable, then we have the Euler product expansion

\overline{D} := ∑_{\begin{matrix}n_{1},...,n_{r}=1 \\ gcd(n_{i},n_{j})=1 (i≠j) \end{matrix}}^{∞} \frac{f_{1}(n_{1}) ⋯ f_{r}(n_{r})}{n_{1}^{s_{1}} ⋯ n_{r}^{s_{r}}}

= ∑_{n_{1},...,n_{r}=1}^{∞} \frac{f_{1}(n_{1}) ⋯ f_{r}(n_{r})ϑ(n_{1}, …, n_{r})}{n_{1}^{s_{1}} ⋯ n_{r}^{s_{r}}}

= ∏_{p} ∑_{a_{1},...,a_{r}=0}^{∞} \frac{f_{1}(p^{a_{1}}) ⋯ f_{r}(p^{a_{r}})ϑ(p^{a_{1}}, …, p^{a_{r}})}{p^{a_{1}s_{1}+···+a_{r}s_{r}}}

= ∏_{p}(1 + ∑_{i=1}^{r} ∑_{a_{i}=1}^{∞} \frac{f_{i}(p^{a_{i}})}{p^{a_{i}s_{i}}})

= ∏_{p}(1 + ∑_{i=1}^{r}(∏_{j=1}^{t}(1 − \frac{g_{ij}(p)}{p^{s_{i}}})^{−1} − 1))

by using (8). We deduce that

\overline{D} = ∏_{p} ∏_{i=1}^{r} ∏_{j=1}^{t}(1 − \frac{g_{ij}(p)}{p^{s_{i}}})^{−1} ∏_{p}((1 − r) ∏_{i=1}^{r} ∏_{j=1}^{t}(1 − \frac{g_{ij}(p)}{p^{s_{i}}})

+ ∑_{i=1}^{r} ∏_{\begin{matrix}k=1 \\ k≠i \end{matrix}}^{r} ∏_{j=1}^{t}(1 − \frac{g_{kj}(p)}{p^{s_{k}}}))

= ∏_{i=1}^{r} D(f_{i}, s_{i}) ∏_{p} K(p),  (14)

say. Let x_{ij} = g_{ij}(p)p^{−s_{i}} (1 ≤ i ≤ r, 1 ≤ j ≤ t). Then by (4),

∏_{j=1}^{t}(1 − \frac{g_{ij}(p)}{p^{s_{i}}}) = ∏_{j=1}^{t}(1 − x_{ij}) = 1 − ∑_{j=1}^{t}(−1)^{j−1}e_{j}(x_{i1}, …, x_{it}) = 1 − y_{i},

with 1 ≤ i ≤ r, where

y_{i} := ∑_{j=1}^{t}(−1)^{j−1}e_{j}(x_{i1}, …, x_{it})

= ∑_{j=1}^{t}(−1)^{j−1} ∑_{1≤ℓ_{1}<···<ℓ_{j}≤r} x_{iℓ_{1}} ⋯ x_{iℓ_{j}}

= ∑_{j=1}^{t} \frac{(−1)^{j−1}}{p^{js_{i}}} ∑_{1≤ℓ_{1}<···<ℓ_{j}≤r} g_{iℓ_{1}}(p) ⋯ g_{iℓ_{j}}(p)

= ∑_{j=1}^{t} \frac{(−1)^{j−1}}{p^{js_{i}}} G_{ij}(p).  (15)

Therefore, by applying Lemma 1 for y_{1}, …, y_{r} we obtain that the expression K(p) under the product ∏_{p} in (14) is

K(p) = (1 − r) ∏_{i=1}^{r}(1 − y_{i}) + ∑_{i=1}^{r} ∏_{\begin{matrix}k=1 \\ k≠i \end{matrix}}^{r}(1 − y_{k})

= 1 + ∑_{i=2}^{r}(−1)^{i−1}(i − 1)e_{i}(y_{1}, …, y_{r})

= 1 + ∑_{i=2}^{r}(−1)^{i−1}(i − 1) ∑_{1≤ℓ_{1}<···<ℓ_{i}≤r} y_{ℓ_{1}} ⋯ y_{ℓ_{i}}

= 1 + ∑_{i=2}^{r}(−1)^{i−1}(i − 1) ∑_{1≤ℓ_{1}<···<ℓ_{i}≤r}( ∑_{a_{ℓ_{1}}=1}^{t} \frac{(−1)^{a_{ℓ_{1}}−1}}{p^{a_{ℓ_{1}}s_{ℓ_{1}}}} G_{ia_{ℓ_{1}}}(p)) × ⋯

× ( ∑_{a_{ℓ_{i}}=1}^{t} \frac{(−1)^{a_{ℓ_{i}}−1}}{p^{a_{ℓ_{i}}s_{ℓ_{i}}}} G_{ia_{ℓ_{i}}}(p))

= 1 − ∑_{i=2}^{r}(i − 1) ∑_{1≤ℓ_{1}<···<ℓ_{i}≤r} ∑_{a_{ℓ_{1}},...,a_{ℓ_{i}}=1}^{t} \frac{1}{p^{a_{ℓ_{1}}s_{ℓ_{1}}+···+a_{ℓ_{i}}s_{ℓ_{i}}}} ∏_{m=1}^{i}(−1)^{a_{ℓm}}G_{ia_{ℓm}}(p),

by (15), ending the proof.

Remark 7. Identity (13) shows that under the assumptions of Theorem 6 we have the convolutional identity

f_{1}(n_{1}) ⋯ f_{r}(n_{r})ϑ(n_{1}, …, n_{r}) = ∑_{d_{1}e_{1}=n_{1},...,d_{r}e_{r}=n_{r}} f_{1}(d_{1}) ⋯ f_{r}(d_{r})\overline{F}_{f_{1},...,f_{r}}(e_{1}, …, e_{r}), (16)

where \overline{F}_{f_{1},...,f_{r}} is the multiplicative function defined for prime powers p^{a_{1}}, …, p^{a_{r}} (a_{1}, …, a_{r} ≥ 0, not all zero) by

\overline{F}_{f_{1},...,f_{r}}(p^{a_{1}}, …, p^{a_{r}}) = \begin{cases}(1 − i) ∏_{m=1}^{i}(−1)^{a_{ℓm}}G_{ia_{ℓm}}(p), & if there exists 2 ≤ i ≤ r and there \\ & exist 1 ≤ ℓ_{1}, …, ℓ_{i} ≤ r such that \\ & 1 ≤ a_{ℓ_{1}}, …, a_{ℓ_{i}} ≤ t; \\ 0, & otherwise. \end{cases}

Corollary 8. Let r ≥ 2 and let t_{i} ≥ 2 (1 ≤ i ≤ r) be fixed integers. If s_{i} ∈ ℂ, ℜs_{i} > 1 (1 ≤ i ≤ r), then

∑_{\begin{matrix}n_{1},...,n_{r}=1 \\ gcd(n_{i},n_{j})=1 (i≠j) \end{matrix}}^{∞} \frac{τ_{t_{1}}(n_{1}) ⋯ τ_{t_{r}}(n_{r})}{n_{1}^{s_{1}} ⋯ n_{r}^{s_{r}}} = ζ^{t_{1}}(s_{1}) ⋯ ζ^{t_{r}}(s_{r})\overline{Δ}(τ_{t_{1}}, …, τ_{t_{r}}, s_{1}, …, s_{r}),

where

\overline{Δ}(τ_{t_{1}}, …, τ_{t_{r}}, s_{1}, …, s_{r}) =

∏_{p}(1 − ∑_{i=2}^{r}(i − 1) ∑_{1≤ℓ_{1}<···<ℓ_{i}≤r} ∑_{a_{ℓ_{1}},...,a_{ℓ_{i}}=1}^{t} \frac{1}{p^{a_{ℓ_{1}}s_{ℓ_{1}}+···+a_{ℓ_{i}}s_{ℓ_{i}}}} ∏_{m=1}^{i}(−1)^{a_{ℓm}} \binom{t_{i}}{a_{ℓ_{m}}}). (17)

Proof. Apply Theorem 6 in the case g_{ij}(p) = 1 for 1 ≤ j ≤ t_{i} and g_{ij}(p) = 0 for t_{i} +1 ≤ j ≤ t (1 ≤ i ≤ r).

Corollary 9. Let r ≥ 2. If s_{i} ∈ ℂ, ℜs_{i} > 1 (1 ≤ i ≤ r), then

∑_{\begin{matrix}n_{1},...,n_{r}=1 \\ gcd(n_{i},n_{j})=1 (i≠j) \end{matrix}}^{∞} \frac{τ(n_{1}) ⋯ τ(n_{r})}{n_{1}^{s_{1}} ⋯ n_{r}^{s_{r}}} = ζ^{2}(s_{1}) ⋯ ζ^{2}(s_{r})

× ∏_{p}(1 − ∑_{i=2}^{r}(i − 1) ∑_{1≤ℓ_{1}<···<ℓ_{i}≤r} ∑_{1≤a_{ℓ_{1}},...,a_{ℓ_{i}}≤2} \frac{(−2)^{#\{1≤m≤i: a_{ℓm}=1\}}}{p^{a_{ℓ_{1}}s_{ℓ_{1}}+···+a_{ℓ_{i}}s_{ℓ_{i}}}} ).

Proof. Apply Corollary 8 for t_{i} = 2 (1 ≤ i ≤ r).

For r = 2 this gives (1) and for r = 3 we have

∑_{n_{1},n_{2},n_{3}=1 gcd(n_{1},n_{2})=gcd(n_{1},n_{3})=gcd(n_{2},n_{3})=1}^{∞} \frac{τ(n_{1})τ(n_{2})τ(n_{3})}{n_{1}^{s_{1}}n_{2}^{n_{2}}n_{3}^{s_{3}}} =

ζ^{2}(s_{1})ζ^{2}(s_{2})ζ^{2}(s_{3}) × ∏_{p}(1 − \frac{4}{p^{s_{1}+s_{2}}} + \frac{2}{p^{2s_{1}+s_{2}}} + \frac{2}{p^{s_{1}+2s_{2}}} − \frac{1}{p^{2s_{1}+2s_{2}}} − \frac{4}{p^{s_{1}+s_{3}}} + \frac{2}{p^{2s_{1}+s_{3}}}

+\frac{2}{p^{s_{1}+2s_{3}}} − \frac{1}{p^{2s_{1}+2s_{3}}} − \frac{4}{p^{s_{2}+s_{3}}} + \frac{2}{p^{2s_{2}+s_{3}}} + \frac{2}{p^{s_{2}+2s_{3}}} − \frac{1}{p^{2s_{2}+2s_{3}}} + \frac{16}{p^{s_{1}+s_{2}+s_{3}}} − \frac{8}{p^{2s_{1}+s_{2}+s_{3}}}

−\frac{8}{p^{s_{1}+2s_{2}+s_{3}}} − \frac{8}{p^{s_{1}+s_{2}+2s_{3}}} + \frac{4}{p^{2s_{1}+2s_{2}+s_{3}}} +\frac{4}{p^{2s_{1}+s_{2}+2s_{3}}} + \frac{4}{p^{s_{1}+2s_{2}+2s_{3}}} − \frac{2}{p^{2s_{1}+2s_{2}+2s_{3}}} ).(18)

If we compare the infinite product (18) to (12), then we can see that in (12) we only have exponents of p of form a_{1}s_{1} + a_{2}s_{2} + a_{3}s_{3} with 1 ≤ a_{1}, a_{2}, a_{3} ≤ 2, while in (18) the exponents of p are a_{1}s_{1} + a_{2}s_{2} + a_{3}s_{3} with 0 ≤ a_{1}, a_{2}, a_{3} ≤ 2 and with at least two nonzero values a_{1}, a_{2}, a_{3}. Similar in the general case, according to Theorems 2 and 6.

It is possible to derive a common generalization of Theorems 2 and 6 by considering k-wise relatively prime integers. Let r ≥ k ≥ 2 be fixed integers. The positive integers n_{1}, …, n_{r} are called k-wise relatively prime if any k of them are relatively prime, that is, gcd(n_{i_{1}}, …, n_{i_{k}}) = 1 for every 1 ≤ i_{1} < ⋯ < i_{k} ≤ r. In particular, in the case k = 2 the integers are pairwise relatively prime and for k = r they are mutually relatively prime. Let ϱ_{k} denote the characteristic function of the set of r-tuples of positive integers with k-wise relatively prime components. Hence, ϱ_{r} = ϱ and ϱ_{2} = ϑ, with our previous notation.

Here we confine ourselves to the case t = 1, that is, the functions f_{1}, …, f_{r} are completely multiplicative.

Theorem 10. Let r ≥ k ≥ 2 and let f_{1}, …, f_{r} : ℕ → ℂ be completely multiplicative functions. Then

∑_{n_{1},...,n_{r}=1}^{∞} \frac{f_{1}(n_{1}) ⋯ f_{r}(n_{r})ϱ_{k}(n_{1}, …, n_{r})}{n_{1}^{s_{1}} ⋯ n_{r}^{s_{r}}} = D(f_{1}, s_{1}) ⋯ D(f_{r}, s_{r})

× ∏_{p}(1 − ∑_{i=k}^{r}(−1)^{i−k}\binom{i − 1}{k − 1} ∑_{1≤ℓ_{1}<···<ℓ_{i}≤r} \frac{f_{ℓ_{1}}(p) ⋯ f_{ℓ_{i}}(p)}{p^{s_{ℓ_{1}}+···+s_{ℓ_{i}}}}),

Proof. For fixed k the function ϱ_{k} is multiplicative. Also, for prime powers p^{a_{1}}, …, p^{a_{r}} (a_{1} …, a_{r} ≥ 0) we have ϱ_{k}(p^{a_{1}}, …, p^{a_{r}}) = 1 if and only there are at most k − 1 values a_{i} ≥ 1. Now the proof is similar to the proofs of Theorems 2 and 6. In the case f_{1}(n) = ⋯ = f_{r}(n) = 1 (n ∈ ℕ) this result and its detailed proof are given in [6, Th. 2.1].

4 Related asymptotic formulas

The above identities can be used to obtain asymptotic formulas with remainder terms for certain related sums. As examples, we point out the following formulas.

Theorem 11. Let r ≥ 2 and let t_{i} ≥ 2 (1 ≤ i ≤ r) be fixed integers. Then for every ε > 0,

∑_{\begin{matrix}n_{1},...,n_{r}≤x \\ gcd(n_{1},...,n_{r})=1 \end{matrix}} τ_{t_{1}}(n_{1}) ⋯ τ_{t_{r}}(n_{r}) = x^{r}Q(log x) + O(x^{r−1+max_{1≤i≤r} ϑ_{t_{i}}+ε}),

where Q(u) is a polynomial in u of degree t_{1} + ⋯ + t_{r} − r having the leading coefficient

\frac{Δ(τ_{t_{1}}, …, τ_{t_{r}}, 1, …, 1)}{(t_{1} − 1)! ⋯ (t_{r} − 1)!},

where Δ(τ_{t_{1}}, …, τ_{t_{r}}, 1, …, 1) is obtained from (11) for s_{1} = ⋯ = s_{r} = 1, and ϑ_{t_{i}} are the exponents in the Piltz divisor problems for τ_{t_{i}}, namely

∑_{n≤x} τ_{t_{i}}(n) = xP_{t_{i}}(log x) + O(x^{ϑ_{t_{i}}+ε}),  (19)

with some polynomials P_{t_{i}}(u) in u of degree t_{i} − 1 having the leading coefficients 1/(t_{i} − 1)! (1 ≤ i ≤ r).

Proof. We have, according to the convolutional identity (10),

∑_{\begin{matrix}n_{1},...,n_{r}≤x \\ gcd(n_{1},...,n_{r})=1 \end{matrix}} τ_{t_{1}}(n_{1}) ⋯ τ_{t_{r}}(n_{r}) = ∑_{d_{1}e_{1}=n_{1}≤x,...,d_{r}e_{r}=n_{r}≤x} τ_{t_{1}}(d_{1}) ⋯ τ_{t_{r}}(d_{r})F_{τ_{t_{1}},...,τ_{tr}} (e_{1}, …, e_{r})

= ∑_{e_{1},...,e_{r}≤x} F_{τ_{t_{1}},...,τ_{tr}} (e_{1}, …, e_{r}) ∑_{d_{1}≤x/e_{1}} τ_{t_{1}}(d_{1}) ⋯ ∑_{d_{r}≤x/e_{r}} τ_{t_{r}}(d_{r}).

Now by using formulas (19) and the fact that the infinite product Δ(τ_{t_{1}}, …, τ_{t_{r}}, s_{1}, …, s_{r}) given by (11) is absolutely convergent provided that s_{i} ∈ ℂ, ℜs_{i} > 0 (1 ≤ i ≤ r), ℜ(s_{1} + ⋯ + s_{r}) > 1, we obtain the desired formula. For the details see the proof of [7, Th. 3.3], which is a generalization of the present result.

Corollary 12. Let r ≥ 2. Then for every ε > 0,

∑_{\begin{matrix}n_{1},...,n_{r}≤x \\ gcd(n_{1},...,n_{r})=1 \end{matrix}} τ(n_{1}) ⋯ τ(n_{r}) = x^{r}T (log x) + O(x^{r−1+θ+ε})

where T (u) is a polynomial in u of degree r having the leading coefficient K_{r}, where

K_{r} = ∏_{p}(1 − (\frac{2p − 1}{p^{2}})^{r}) = ∏_{p}(1 − ∑_{i=0}^{r}(−1)^{i}\binom{r}{i}\frac{2^{r−i}}{p^{r+i}} );

in particular,

K_{2} = ∏_{p}(1 − \frac{4}{p^{2}} + \frac{4}{p^{3}} − \frac{1}{p^{4}} ),  (20)

K_{3} = ∏_{p}(1 − \frac{8}{p^{3}} + \frac{12}{p^{4}} − \frac{6}{p^{5}} + \frac{1}{p^{6}} ),

and θ is the exponent in Dirichlet’s divisor problem.

Proof. Apply Theorem 11 in the case t_{i} = 2 (≤ i ≤ r). The representation of K_{r} follows from (9) for g_{ij}(p) = 1 (1 ≤ i ≤ r, 1 ≤ j ≤ 2).

We note that in the case r = 2 this result has been proved in [3, Lemma 3.3] by analytic methods, with a weaker error term.

Theorem 13. Let r ≥ 2 and let t_{i} ≥ 2 (1 ≤ i ≤ r) be fixed integers. Then for every ε > 0,

∑_{\begin{matrix}n_{1},...,n_{r}≤x \\ gcd(n_{i},n_{j})=1 (i≠j) \end{matrix}} τ_{t_{1}}(n_{1}) ⋯ τ_{t_{r}}(n_{r}) = x^{r}\overline{Q}(log x) + O(x^{r−1+max_{1≤i≤r} ϑ_{t_{i}}+ε}),

where \overline{Q}(u) is a polynomial in u of degree t_{1} + ⋯ + t_{r} − r having the leading coefficient

\frac{\overline{Δ}(τ_{t_{1}}, …, τ_{t_{r}}, 1, …, 1)}{(t_{1} − 1)! ⋯ (t_{r} − 1)!},

where \overline{Δ}(τ_{t_{1}}, …, τ_{t_{r}}, 1, …, 1) is obtained from (17) for s_{1} = ⋯ = s_{r} = 1, and ϑ_{t_{i}} are the exponents in the Piltz divisor problems for τ_{t_{i}} (1 ≤ i ≤ r).

Proof. Similar to the proof of Theorem 11. By the convolutional identity (16) we have

∑_{\begin{matrix}n_{1},...,n_{r}≤x \\ gcd(n_{i},n_{j})=1 (i≠j) \end{matrix}} τ_{t_{1}}(n_{1}) ⋯ τ_{t_{r}}(n_{r}) = ∑_{d_{1}e_{1}=n_{1}≤x,...,d_{r}e_{r}=n_{r}≤x} τ_{t_{1}}(d_{1}) ⋯ τ_{t_{r}}(d_{r})\overline{F}_{τ_{t_{1}},...,τ_{tr}} (e_{1}, …, e_{r}),

= ∑_{e_{1},...,e_{r}≤x} \overline{F}_{τ_{t_{1}},...,τ_{tr}} (e_{1}, …, e_{r}) ∑_{d_{1}≤x/e_{1}} τ_{t_{1}}(d_{1}) × ⋯

× ∑_{d_{r}≤x/e_{r}} τ_{t_{r}}(d_{r}).

Now use formulas (19) and the fact that the infinite product \overline{Δ}(τ_{t_{1}}, …, τ_{t_{r}}, s_{1}, …, s_{r}) given by (17) is absolutely convergent provided that s_{i} ∈ ℂ, ℜs_{i} > 0 (1 ≤ i ≤ r), ℜ(s_{i}+s_{j}) > 1 (1 ≤ i < j ≤ r). This is also a special case of [7, Th. 3.3].

Corollary 14. Let r ≥ 2. Then for every ε > 0,

∑_{\begin{matrix}n_{1},...,n_{r}≤x \\ gcd(n_{i},n_{j})=1 (i≠j) \end{matrix}} τ(n_{1}) ⋯ τ(n_{r}) = x^{r}\overline{T} (log x) + O(x^{r−1+θ+ε})

where \overline{T} (u) is a polynomial in u of degree r having the leading coefficient \overline{K}_{r}, where

\overline{K}_{r} = ∏_{p}(1 − \frac{1}{p})^{2(r−1)}(1 + \frac{(r − 1)(2p − 1)}{p^{2}}),

in particular, \overline{K}_{2} = K_{2} given by (20),

\overline{K}_{3} = ∏_{p}(1 − \frac{12}{p^{2}} + \frac{28}{p^{3}} − \frac{27}{p^{4}} + \frac{12}{p^{5}} − \frac{2}{p^{6}} ),

and θ is the exponent in Dirichlet’s divisor problem.

Proof. Apply Theorem 13 in the case t_{i} = 2 (≤ i ≤ r). The representation of \overline{K}_{r} follows from (14) for g_{ij}(p) = 1 (1 ≤ i ≤ r, 1 ≤ j ≤ 2).

References

[1] T. M. Apostol, Introduction to Analytic Number Theory, Springer, 1976.

[2] H. Delange, On some sets of pairs of positive integers, J. Number Theory 1 (1969), 261–279.

[3] W. G. Nowak and L. Tóth, On the average number of subgroups of the group ℤ_{m} × ℤ_{n}, Int. J. Number Theory 10 (2014), 363–374.

[4] N. Tahmi and A. Derbal, Some multiple Dirichlet series of completely multiplicative arithmetic functions, Notes Number Theory Discrete Math. 28 (2022), 603–616.

[5] L. Tóth, Multiplicative arithmetic functions of several variables: a survey, in Mathematics Without Boundaries, Surveys in Pure Mathematics, Springer, 2014, pp. 483–514.

[6] L. Tóth, Counting r-tuples of positive integers with k-wise relatively prime components, J. Number Theory 166 (2016), 105–116.

[7] L. Tóth and W. Zhai, On multivariable averages of divisor functions, J. Number Theory 192 (2018), 251–269.

2020 Mathematics Subject Classification: Primary 11A25; Secondary 11A05, 11N37. Keywords: relatively prime integers, pairwise relatively prime integers, arithmetic function of several variables, multiple Dirichlet series, multiplicative function, completely multiplicative function, asymptotic formula.

(Concerned with sequence \underline{A000005}.)

Received December 29 2022; revised version received January 23 2023. Published in Journal of Integer Sequences, February 2 2023.

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