Sums of Products of Generalized Ramanujan Sums

Soichi Ikeda Department of Mathematics

Shibaura Institute of Technology 307 Fukasaku, Minuma-ku, Saitama 337-8570

Japan

sikeda@sic.shibaura-it.ac.jp

Isao Kiuchi

Department of Mathematical Sciences

Faculty of Science Yamaguchi University

Yoshida 1677-1 Yamaguchi 753-8512

Japan

kiuchi@yamaguchi-u.ac.jp

Kaneaki Matsuoka Graduate School of Mathematics

Nagoya University Furocho Chikusaku Nagoya 464-8602

Japan

m10041v@math.nagoya-u.ac.jp

Abstract

We consider weighted averages for the products t_{k_{1}}^{(1)}(j) ⋯ t_{k_{n}}^{(n)}(j) of generalized Ramanujan sums t_{k_{i}}^{(i)}(j) = ∑_{d| gcd(k_{i},j)} f_{i}(d)g_{i}(k_{i}/d)h_{i}(j/d) with any arithmetical functions f_{i}, g_{i} and h_{i} (i = 1, …, n), and derive formulas for several weighted averages with weights concerning completely multiplicative functions, completely additive functions, and others.

1 Introduction

Let gcd(k, j) be the greatest common divisor of the positive integer k and the integer j, and let K = lcm(k_{1}, …, k_{n}) be the least common multiple of n-tuple positive integers k_{1}, …, k_{n}. The function c_{k}(j), as usual, denotes the Ramanujan sum defined as the sum of the m-th powers of the primitive k-th roots of unity, namely,

c_{k}(j) = ∑_{{1≤m≤k \atop gcd(m,k)=1}} exp(2πi\frac{mj}{k} ),

which can be expressed as the well-known identity

∑_{d| gcd(k,j)} dμ(\frac{k}{d})

with the Möbius function μ. Anderson and Apostol [5] (also see [6, 7, 14, 17]) first introduced the sum

s_{k}(j) = ∑_{d| gcd(k,j)} f(d)g (\frac{k}{d})

with any arithmetical functions f and g, which is a generalization of Ramanujan’s sum. This function is said to be the Anderson–Apostol sum. Now, let f be a completely multiplicative function, and let g(k) = μ(k)u(k) with u a multiplicative function. Assume that f(p) ≠ 0 and f(p) ≠ u(p) for all primes p, and let F (k) = (f ∗ g)(k). Anderson and Apostol [5] derived the identity s_{k}(j) = F (k)μ (m) u (m)/F (m) with m = k/ gcd(k, j). This is said to be Hölder’s identity for s_{k}(j), which has been considered by many mathematicians and physicists. For any fixed positive integer r, using the properties of the sum

∑_{j=1}^{k} j^{r}c_{k}(j),  (1)

Alkan [1, 3] derived exact formulas involving Euler’s and Jordan’s functions for averages of the special values of L-functions. Tóth [18] showed a simpler proof of (1) and established some identities for other weighted averages of Ramanujan’s sum with weights concerning logarithms, values of arithmetic functions for gcd’s, the gamma function, the Bernoulli polynomials, and binomial coefficients. In a recent paper, Kiuchi, Minamide and Ueda [11] gave a generalization of some identities due to Tóth [18]; they showed that

\frac{1}{k^{r}} ∑_{j=1}^{k} j^{r}s_{k}(j) = \frac{1}{2}(f ∗ g)(k) + \frac{1}{r + 1} ∑_{m=0}^{⌊r/2⌋} \binom{r + 1}{2m}B_{2m}(f ∗ id_{1−2m} · g)(k)  (2)

for any fixed positive integer r,

∑_{j=1}^{k} s_{k}(j) log j = (f · log ∗g · id)(k) + (f ∗ g · Log)(k),  (3)

∑_{j=1}^{k} s_{k}(j) log Γ (\frac{j}{k} ) = log \sqrt{2π} \{(f ∗ g · id)(k) − (f ∗ g)(k)\} − \frac{1}{2}(f ∗ g · log)(k), (4)

\frac{1}{2^{k}} ∑_{j=0}^{k} \binom{k}{j}s_{k}(j) = ∑_{d|k} \frac{f(d)}{d}g (\frac{k}{d}) ∑_{l=1}^{d}(−1)^{\frac{lk}{d}} cos^{k} \frac{lπ}{d}  (5)

and

∑_{j=0}^{k−1} B_{m} (\frac{j}{k} ) s_{k}(j) = \frac{B_{m}}{k^{k−1}} (id_{m−1} · f ∗ g)(k),  (6)

where the function Log d is given by log(d!) and Γ denotes the gamma function. Here, B_{m}(x)

[7] (also see [8, 9]) denotes the Bernoulli polynomials defined by the expansion

\frac{ze^{xz}}{e^{z} − 1} = ∑_{m=0}^{∞} B_{m}(x)\frac{z^{m}}{m!}

where |z| < 2π. The number B_{m} is the Bernoulli number given by B_{m}(0). The expressions in (2)–(6) give a generalization of some interesting identities by Tóth [12] and Alkan [1, 2, 3, 4]. The function

E(k_{1}, …, k_{n}) = \frac{1}{K} ∑_{j=1}^{K} c_{k_{1}}(j) ⋯ c_{k_{n}}(j)  (7)

of the product c_{k_{1}}(j) ⋯ c_{k_{n}}(j) of Ramanujan’s sums for n variables was investigated in the studies of Liskovets [12] and Tóth [20], which has some interesting formulas for combinatorial and topological applications. The expression (7) has been introduced by Mednykh and Nedela [15] to handle certain problems of enumerative combinatorics. They showed that all the values E(k_{1}, …, k_{n}) are nonnegative integers. Furthermore, Tóth derived that two interesting representations [20, Propositions 3 and 9], [19, Corollary 4] for E hold:

E(k_{1}, …, k_{n}) = ∑_{d_{1}|k_{1},...,d_{n}|k_{n}} \frac{d_{1} ⋯ d_{n}}{lcm(d_{1}, …, d_{n})}μ (\frac{k_{1}}{d_{1}} ) ⋯ μ (\frac{k_{n}}{d_{n}} )  (8)

and

E(k_{1}, …, k_{n}) = \frac{1}{K} ∑_{d|K} c_{k_{1}}(d) ⋯ c_{k_{n}}(d)ϕ (\frac{K}{d} ) ,  (9)

where ϕ is the Euler totient function. He also noted that all values for E are nonnegative integers and that E is multiplicative as a function of several variables. Let Ẽ denote a generalization of E defined by

Ẽ(k_{1}, …, k_{n}) = \frac{1}{K} ∑_{j=1}^{K} s_{k_{1}}(j) ⋯ s_{k_{n}}(j).  (10)

Then Tóth [20, Proposition 19] deduced that two formulas for Ẽ hold:

Ẽ(k_{1}, …, k_{n}) = ∑_{d_{1}|k_{1},...,d_{n}|k_{n}} \frac{f(d_{1}) ⋯ f(d_{n})}{lcm(d_{1}, …, d_{n})}g (\frac{k_{1}}{d_{1}} ) ⋯ g (\frac{k_{n}}{d_{n}} )  (11)

and

Ẽ(k_{1}, …, k_{n}) = \frac{1}{K} ∑_{d|K} s_{k_{1}}(d) ⋯ s_{k_{n}}(d)ϕ (\frac{K}{d} ) .  (12)

He also showed that if f and g are multiplicative functions, then (10) is multiplicative as a function of several variables. The weighted average of the products of Ramanujan’s sum with weight concerning the function id_{r} for any fixed positive integer r defined by

S_{r}(k_{1}, …, k_{n}) = \frac{1}{K^{r+1}} ∑_{j=1}^{K} j^{r}c_{k_{1}}(j) ⋯ c_{k_{n}}(j)  (13)

was considered by Tóth [18, Proposition 7], who derived the following identity

S_{r}(k_{1}, …, k_{n}) = \frac{ϕ(k_{1}) ⋯ ϕ(k_{n})}{2K}  (14)

+ \frac{1}{r + 1} ∑_{m=0}^{⌊ \frac{r}{2} ⌋} \binom{r + 1}{2m}\frac{B_{2m}}{K^{2m}} ∑_{d_{1}|k_{1},...,d_{n}|k_{n}} \frac{d_{1} ⋯ d_{n}}{lcm(d_{1}, …, d_{n})^{1−2m}} μ (\frac{k_{1}}{d_{1}} ) ⋯ μ (\frac{k_{n}}{d_{n}} ) .

He presented a multivariable generalization of (1) connected to the orbicyclic arithmetic function, discussed by Liskovets [12] and Tóth [20]. Substituting r = 1 in (14), it follows that

S_{1}(k_{1}, …, k_{n}) = \frac{ϕ(k_{1}) ⋯ ϕ(k_{n})}{2K} + \frac{E(k_{1}, …, k_{n})}{2},

which was given by Tóth [18, Corollary 2].

For any arithmetical functions f_{i}, g_{i} and h_{i} (i = 1, 2, …, n), we define s_{k_{i}}^{(i)}(j) and t_{k_{i}}^{(i)}(j) by

s_{k_{i}}^{(i)}(j) = ∑_{d| gcd(k_{i},j)} f_{i}(d)g_{i} (\frac{k_{i}}{d} ) and t_{k_{i}}^{(i)}(j) = ∑_{d| gcd(k_{i},j)} f_{i}(d)g_{i} (\frac{k_{i}}{d} ) h_{i} (\frac{j}{d}) ,

respectively.

The first aim of this study is to derive some identities for weighted averages of the product t_{k_{1}}^{(1)}(j) ⋯ t_{k_{n}}^{(n)}(j) with weight function w_{r}, completely multiplicative, completely additive, and others, for any fixed positive integer r, namely

U_{r}(k_{1}, …, k_{n}) = ∑_{j=1}^{K} w_{r}(j)t_{k_{1}}^{(1)}(j) ⋯ t_{k_{n}}^{(n)}(j).  (15)

This sum is a generalization of some of the identities for weighted averages mentioned by Alkan [2], Liskovets [12], Tóth [18, 20] and Kiuchi, Minamide and Ueda [11]. To our knowledge, we derive some new and useful formulas in Theorems 1, 6 and 10. This study then aims to establish some identities for the weighted averages of the product s_{k_{1}}^{(1)}(j) ⋯ s_{k_{n}}^{(n)}(j) with weights concerning the gamma function, binomial coefficients, and Bernoulli polynomials, which provide a generalization of some interesting identities discussed by Tóth [18] and Kiuchi, Minamide and Ueda [11].

Notations. We use the following notation throughout this paper. For any positive integer n, the functions id and id_{q} as well as the unit function 1 are given by id(n) = n, id_{q}(n) = n^{q} for any real number q and 1(n) = 1, respectively. The symbols ∗ and · denote the Dirichlet convolution and the ordinary product of arithmetical functions, respectively. The function ϕ_{s}(n) defines the Jordan function by (id_{s} ∗ μ)(n) for any real number s.

2 Some formulas for U_{r}(k_{1}, …, k_{n})

We shall evaluate the function U_{r}(k_{1}, …, k_{n}). The weight function w_{r} of U_{r}(k_{1}, …, k_{n}) only deals with completely multiplicative functions and completely additive functions, and we introduce a simple proof of (15) and some useful formulas in Theorems 1 and 6. Moreover, we shall derive some identities of the weighted averages of the product s_{k_{1}}^{(1)}(j) ⋯ s_{k_{n}}^{(n)}(j) with weights concerning the gamma function, binomial coefficients, and the Bernoulli polynomials.

Theorem 1. Let f_{1}, …, f_{n}, g_{1}, …, g_{n}, and h_{1}, …, h_{n} denote any arithmetical functions, and let

U_{r}(k_{1}, …, k_{n})

= ∑_{j=1}^{K} w_{r}(j) ∑_{d_{1}| gcd(k_{1},j)} f_{1}(d_{1})g_{1} (\frac{k_{1}}{d_{1}} ) h_{1} (\frac{j}{d_{1}} ) ⋯ ∑_{d_{n}| gcd(k_{n},j)} f_{n}(d_{n})g_{n} (\frac{k_{n}}{d_{n}} ) h_{n} (\frac{j}{d_{n}} ) .

If w_{r} is a completely multiplicative function, we have

U_{r}(k_{1}, …, k_{n})  (16)

= ∑_{d_{1}|k_{1},...,d_{n}|k_{n}} f_{1}(d_{1})g_{1} (\frac{k_{1}}{d_{1}} ) ⋯ f_{n}(d_{n})g_{n} (\frac{k_{n}}{d_{n}} ) w_{r}(lcm(d_{1}, …, d_{n}))×

× ∑_{l=1}^{\frac{K}{lcm(d_{1},...,dn)}} w_{r}(l)h_{1} (\frac{lcm(d_{1}, …, d_{n})}{d_{1}}l) ⋯ h_{n} (\frac{lcm(d_{1}, …, d_{n})}{d_{n}}l) ,

and if in addition, h_{1}, …, h_{n} are completely multiplicative functions, then

U_{r}(k_{1}, …, k_{n})  (17)

= ∑_{d_{1}|k_{1},...,d_{n}|k_{n}} f_{1}(d_{1})g_{1} (\frac{k_{1}}{d_{1}} ) h_{1} (\frac{lcm(d_{1}, …, d_{n})}{d_{1}}) ⋯ f_{n}(d_{n})g_{n} (\frac{k_{n}}{d_{n}} ) ×

× h_{n} (\frac{lcm(d_{1}, …, d_{n})}{d_{n}}) w_{r} (lcm(d_{1}, …, d_{n})) ∑_{l=1}^{\frac{K}{lcm(d_{1},...,dn)}} w_{r}(l)h_{1} (l) ⋯ h_{n} (l) .

If w_{r} is a completely additive function, we have

U_{r}(k_{1}, …, k_{n})  (18)

= ∑_{d_{1}|k_{1},...,d_{n}|k_{n}} f_{1}(d_{1})g_{1} (\frac{k_{1}}{d_{1}} ) ⋯ f_{n}(d_{n})g_{n} (\frac{k_{n}}{d_{n}} ) w_{r}(lcm(d_{1}, …, d_{n}))×

× ∑_{l=1}^{\frac{K}{lcm(d_{1},...,dn)}} h_{1} (\frac{lcm(d_{1}, …, d_{n})}{d_{1}}l) ⋯ h_{n} (\frac{lcm(d_{1}, …, d_{n})}{d_{n}}l)

+ ∑_{d_{1}|k_{1},...,d_{n}|k_{n}} f_{1}(d_{1})g_{1} (\frac{k_{1}}{d_{1}} ) ⋯ f_{n}(d_{n})g_{n} (\frac{k_{n}}{d_{n}} ) ×

× ∑_{l=1}^{\frac{K}{lcm(d_{1},...,dn)}} w_{r}(l)h_{1} (\frac{lcm(d_{1}, …, d_{n})}{d_{1}}l) ⋯ h_{n} (\frac{lcm(d_{1}, …, d_{n})}{d_{n}}l) ,

and if in addition, h_{1}, …, h_{n} are completely multiplicative functions, then

U_{r}(k_{1}, …, k_{n})  (19)

= ∑_{d_{1}|k_{1},...,d_{n}|k_{n}} f_{1}(d_{1})g_{1} (\frac{k_{1}}{d_{1}} ) h_{1} (\frac{lcm(d_{1}, …, d_{n})}{d_{1}}) ⋯ f_{n}(d_{n})g_{n} (\frac{k_{n}}{d_{n}} ) h_{n} (\frac{lcm(d_{1}, …, d_{n})}{d_{n}}) ×

× w_{r}(lcm(d_{1}, …, d_{n})) ∑_{l=1}^{\frac{K}{lcm(d_{1},...,dn)}} h_{1} (l) ⋯ h_{n} (l)

+ ∑_{d_{1}|k_{1},...,d_{n}|k_{n}} f_{1}(d_{1})g_{1} (\frac{k_{1}}{d_{1}} ) h_{1} (\frac{lcm(d_{1}, …, d_{n})}{d_{1}}) ⋯ f_{n}(d_{n})g_{n} (\frac{k_{n}}{d_{n}} ) h_{n} (\frac{lcm(d_{1}, …, d_{n})}{d_{n}}) ×

× ∑_{l=1}^{\frac{K}{lcm(d_{1},...,dn)}} w_{r}(l)h_{1} (l) ⋯ h_{n} (l) .

Remark 2. The formulas (16) and (17) immediately imply a generalization of the two formulas (8) and (11). We substitute w_{r} = 1, f_{1} = ⋯ = f_{n} = f, g_{1} = ⋯ = g_{n} = g, and h_{1} = ⋯ = h_{n} = 1 in (16) to obtain the formula (11).

The formulas (16)–(19) give an analogue and a generalization of result of Tóth [22, Proposition 1]. For any positive integer k, Kiuchi, Minamide and Ueda [11] recently showed that if w_{r} is a completely multiplicative function, then the identity

∑_{j=1}^{k} w_{r}(j)t_{k}(j) = (f · w_{r} ∗ g · W)(k)  (20)

holds with W (d) = ∑_{m=1}^{d} w_{r}(m)h(m), and if w_{r} is a completely additive function, then the identity

∑_{j=1}^{k} w_{r}(j)t_{k}(j) = (f · w_{r} ∗ g · H)(k) + (f ∗ g · W)(k)  (21)

holds with H(d) = ∑_{m=1}^{d} h(m). Thus, the formulas (16) and (18) give a generalization of (20) and (21), respectively. Substituting w_{r} = id_{r} and h_{1} = ⋯ = h_{n} = 1 in (17), w_{r} = id_{r} and g_{1} = ⋯ = g_{n} = 1 in (16), and w_{r} = id_{r} and f_{1} = ⋯ = f_{1} = 1 in (16), we derive the following formulas (22), (23), and (24), respectively.

Corollary 3. Let the notation be as above. Then we have

\frac{1}{K^{r+1}} ∑_{j=1}^{K} j^{r}s_{k_{1}}^{(1)}(j) ⋯ s_{k_{n}}^{(n)}(j)  (22)

= \frac{1}{2K} (f_{1} ∗ g_{1})(k_{1}) ⋯ (f_{n} ∗ g_{n})(k_{n})

+ \frac{1}{r + 1} ∑_{m=0}^{⌊r/2⌋} \binom{r + 1}{2m}\frac{B_{2m}}{K^{2m}} ∑_{d_{1}|k_{1},...,d_{n}|k_{n}} \frac{f_{1}(d_{1}) ⋯ f_{n}(d_{n})}{lcm(d_{1}, …, d_{n})^{1−2m}} g_{1} (\frac{k_{1}}{d_{1}} ) ⋯ g_{n} (\frac{k_{n}}{d_{n}} ) ,

∑_{j=1}^{K} j^{r} ∑_{d_{1}| gcd(k_{1},j)} f_{1}(d_{1})h_{1} (\frac{j}{d_{1}} ) ⋯ ∑_{d_{n}| gcd(k_{n},j)} f_{n}(d_{n})h_{n} (\frac{j}{d_{n}} )  (23)

= ∑_{d_{1}|k_{1},...,d_{n}|k_{n}} f_{1}(d_{1}) ⋯ f_{n}(d_{n}) lcm(d_{1}, …, d_{n})^{r}×

× ∑_{l=1}^{\frac{K}{lcm(d_{1},...,dn)}} l^{r}h_{1} (\frac{lcm(d_{1}, …, d_{n})}{d_{1}}l) ⋯ h_{n} (\frac{lcm(d_{1}, …, d_{n})}{d_{n}}l)

and

∑_{j=1}^{K} j^{r} ∑_{d_{1}| gcd(k_{1},j)} g_{1} (\frac{k_{1}}{d_{1}} ) h_{1} (\frac{j}{d_{1}} ) ⋯ ∑_{d_{n}| gcd(k_{n},j)} g_{n} (\frac{k_{n}}{d_{n}} ) h_{n} (\frac{j}{d_{n}} )  (24)

= ∑_{d_{1}|k_{1},...,d_{n}|k_{n}} g_{1} (\frac{k_{1}}{d_{1}} ) ⋯ g_{n} (\frac{k_{n}}{d_{n}} ) lcm(d_{1}, …, d_{n})^{r}×

× ∑_{l=1}^{\frac{K}{lcm(d_{1},...,dn)}} l^{r}h_{1} (\frac{lcm(d_{1}, …, d_{n})}{d_{1}}l) ⋯ h_{n} (\frac{lcm(d_{1}, …, d_{n})}{d_{n}}l) .

The formula (22) also gives a generalization of the formula (2). As an application of Corollary 3, we give some formulas for weighted averages of the products of the arithmetical functions of Anderson–Apostol sums.

Example 4. Let the notation be as above. Then we have

\frac{1}{K^{r+1}} ∑_{j=1}^{K} j^{r} gcd(k_{1}, j) ⋯ gcd(k_{n}, j)  (25)

= \frac{k_{1}k_{2} ⋯ k_{n}}{2K} + \frac{1}{r + 1} ∑_{m=0}^{⌊r/2⌋} \binom{r + 1}{2m}\frac{B_{2m}}{K^{2m}} ∑_{d_{1}|k_{1},...,d_{n}|k_{n}} \frac{ϕ(d_{1}) ⋯ ϕ(d_{n})}{lcm(d_{1}, …, d_{n})^{1−2m}} ,

(26)

\frac{1}{K^{2r}} ∑_{j=1}^{K} j^{2r} ∑_{d_{1}| gcd(k_{1},j)} \frac{ϕ(d_{1})}{d_{1}} ⋯ ∑_{d_{1}|(k_{r},j)} \frac{ϕ(d_{r})}{d_{r}}  (27)

= \frac{1}{2} ∑_{d_{1}|k_{1},...,d_{r}|k_{r}} \frac{ϕ(d_{1}) ⋯ ϕ(d_{r})}{d_{1} ⋯ d_{r}}

+ \frac{K}{2r + 1} ∑_{m=0}^{r} \binom{2r + 1}{2m}\frac{B_{2m}}{K^{2m}} ∑_{d_{1}|k_{1},...,d_{r}|k_{r}} \frac{ϕ(d_{1})}{d_{1}} ⋯ \frac{ϕ(d_{r})}{d_{r}} lcm(d_{1}, …, d_{r})^{2m−1},

\frac{1}{K^{r+n+1}} ∑_{j=1}^{K} j^{r+n} ∑_{d_{1}| gcd(k_{1},j)} \frac{f_{1}(d_{1})}{d_{1}} ⋯ ∑_{d_{n}|(k_{n},j)} \frac{f_{n}(d_{n})}{d_{n}}  (28)

= \frac{1}{2K} ∑_{d_{1}|k_{1}} \frac{f_{1}(d_{1})}{d_{1}} ⋯ ∑_{d_{n}|k_{n}} \frac{f_{n}(d_{1})}{d_{n}}

+ \frac{1}{r + n + 1} ∑_{m=0}^{⌊ \frac{r+n}{2} ⌋} \binom{r + n + 1}{2m}\frac{B_{2m}}{K^{2m}} ∑_{d_{1}|k_{1},...,d_{n}|k_{n}} f_{1} (d_{1}) ⋯ f_{n} (d_{n}) \frac{lcm(d_{1}, …, d_{n})^{2m−1}}{d_{1} ⋯ d_{n}} ,

and

\frac{1}{K^{r+n+1}} ∑_{j=1}^{K} j^{r+n} ∑_{d_{1}| gcd(k_{1},j)} g_{1} (\frac{k_{1}}{d_{1}} ) \frac{1}{d_{1}} ⋯ ∑_{d_{n}| gcd(k_{n},j)} g_{n} (\frac{k_{n}}{d_{n}} ) \frac{1}{d_{n}}  (29)

= \frac{1}{2K} (g_{1} ∗ \frac{1}{id}) (k_{1}) ⋯ (g_{n} ∗ \frac{1}{id}) (k_{n})

+ \frac{1}{r + n + 1} ∑_{m=0}^{⌊ \frac{r+n}{2} ⌋} \binom{r + n + 1}{2m}\frac{B_{2m}}{K^{2m}} ∑_{d_{1}|k_{1},...,d_{n}|k_{n}} g_{1} (\frac{k_{1}}{d_{1}} ) ⋯ g_{n} (\frac{k_{n}}{d_{n}} ) \frac{lcm(d_{1}, …, d_{n})^{2m−1}}{d_{1} ⋯ d_{n}} .

We substitute f_{1} = ⋯ = f_{n} = ϕ and g_{1} = ⋯ = g_{n} = 1 in (22) to obtain (25) using (ϕ ∗ 1)(gcd(k_{i}, j)) = gcd(k_{i}, j) (i = 1, …, n). The formula (25) is an analogue of the identity

\frac{1}{K} ∑_{j=1}^{K} gcd(k_{1}, j) ⋯ gcd(k_{n}, j) = ∑_{d_{1}|k_{1},...,d_{n}|k_{n}} \frac{ϕ(d_{1}) ⋯ ϕ(d_{n})}{lcm(d_{1}, …, d_{n})}  (30)

[20, Proposition 12], [21, Corollary 2]. The formula (30) was mentioned by Liskovets [12], and it was considered by Deitmar, Koyama and Kurokawa [10] in a special case, investigating analytic properties of Igusa-type zeta-functions. Using the arguments of elementary probability theory, the explicit formula for the values of (30) derived in [10] was re-proved by Minami [13] for the general case m_{1}, …, m_{n}. We substitute n = r, f_{1} = ⋯ = f_{n} = ϕ and h_{1} = ⋯ = h_{n} = id in (23) and use (51) below to obtain (27). The formulas (23) and (24) imply a generalization of two formulas derived in [11], namely

\frac{1}{k^{r}} ∑_{j=1}^{k} j^{r} ∑_{d| gcd(k,j)} f(d)h (\frac{j}{d}) = ∑_{d|k} f (\frac{k}{d}) \frac{1}{d^{r}} ∑_{l=1}^{d} h(l)l^{r}

and

\frac{1}{k^{r}} ∑_{j=1}^{k} j^{r} ∑_{d| gcd(k,j)} g (\frac{k}{d}) h (\frac{j}{d}) = ∑_{d|k} \frac{g(d)}{d^{r}} ∑_{l=1}^{d} h(l)l^{r},

which follow from (20). Substituting h_{1} = ⋯ = h_{n} = id in (23) and (24), respectively, and using (51) below we easily obtain the two formulas (28) and (29).

We shall evaluate some identities of weighted averages for the product t_{k_{1}}^{(1)} ⋯ t_{k_{n}}^{(n)} of the weight w_{r} with the completely additive function. We set w_{r} = log and substitute h_{1} = ⋯ = h_{n} = 1 in (18), g_{1} = ⋯ = g_{n} = 1 in (18), and f_{1} = ⋯ = f_{n} = 1 in (18) to obtain the following formulas (31), (32), and (33), respectively.

Corollary 5. Let the notation be as above. Then we have

∑_{j=1}^{K} s_{k_{1}}^{(1)}(j) ⋯ s_{k_{n}}^{(n)}(j) log j  (31)

= K ∑_{d_{1}|k_{1},...,d_{n}|k_{n}} f_{1}(d_{1})g_{1} (\frac{k_{1}}{d_{1}} ) ⋯ f_{n}(d_{n})g_{n} (\frac{k_{n}}{d_{n}} ) \frac{log lcm(d_{1}, …, d_{n})}{lcm(d_{1}, …, d_{n})}

+ ∑_{d_{1}|k_{1},...,d_{n}|k_{n}} f_{1}(d_{1})g_{1} (\frac{k_{1}}{d_{1}} ) ⋯ f_{n}(d_{n})g_{n} (\frac{k_{n}}{d_{n}} ) log (\frac{K}{lcm(d_{1}, …, d_{n})})!

∑_{j=1}^{K} log j ∑_{d_{1}| gcd(k_{1},j)} f_{1}(d_{1})h_{1} (\frac{j}{d_{1}} ) ⋯ ∑_{d_{n}| gcd(k_{n},j)} f_{n}(d_{n})h_{n} (\frac{j}{d_{n}} )  (32)

= ∑_{d_{1}|k_{1},...,d_{n}|k_{n}} f_{1}(d_{1}) ⋯ f_{n}(d_{n}) log lcm(d_{1}, …, d_{n})×

× ∑_{l=1}^{\frac{K}{lcm(d_{1},...,dn)}} h_{1} (\frac{lcm(d_{1}, …, d_{n})}{d_{1}}l) ⋯ h_{n} (\frac{lcm(d_{1}, …, d_{n})}{d_{n}}l)

+ ∑_{d_{1}|k_{1},...,d_{n}|k_{n}} f_{1}(d_{1}) ⋯ f_{n}(d_{n}) ∑_{l=1}^{\frac{K}{lcm(d_{1},...,dn)}} h_{1} (\frac{lcm(d_{1}, …, d_{n})}{d_{1}}l) ⋯ h_{1} (\frac{lcm(d_{1}, …, d_{n})}{d_{n}}l) log l

and

∑_{j=1}^{K} log j ∑_{d_{1}| gcd(k_{1},j)} g_{1} (\frac{k_{1}}{d_{1}} ) h_{1} (\frac{j}{d_{1}} ) ⋯ ∑_{d_{n}| gcd(k_{n},j)} g_{n} (\frac{k_{n}}{d_{n}} ) h_{n} (\frac{j}{d_{n}} )  (33)

= ∑_{d_{1}|k_{1},...,d_{n}|k_{n}} g_{1} (\frac{k_{1}}{d_{1}} ) ⋯ g_{n} (\frac{k_{n}}{d_{n}} ) log lcm(d_{1}, …, d_{n})×

× ∑_{l=1}^{\frac{K}{lcm(d_{1},...,dn)}} h_{1} (\frac{lcm(d_{1}, …, d_{n})}{d_{1}}l) ⋯ h_{n} (\frac{lcm(d_{1}, …, d_{n})}{d_{n}}l)

+ ∑_{d_{1}|k_{1},...,d_{n}|k_{n}} g_{1} (\frac{k_{1}}{d_{1}} ) ⋯ g_{n} (\frac{k_{n}}{d_{n}} ) ×

× ∑_{l=1}^{\frac{K}{lcm(d_{1},...,dn)}} h_{1} (\frac{lcm(d_{1}, …, d_{n})}{d_{1}}l) ⋯ h_{n} (\frac{lcm(d_{1}, …, d_{n})}{d_{n}}l) log l.

(31) implies a generalization of (3). (32) and (33) give a generalization of the formulas

∑_{j=1}^{k} log j ∑_{d| gcd(k,j)} f(d)h (\frac{j}{d}) = (f · log ∗H)(k) + (f ∗ I)(k),

and

∑_{j=1}^{k} log j ∑_{d| gcd(k,j)} g (\frac{k}{d}) h (\frac{j}{d}) = (log ∗g · H)(k) + (1 ∗ g · W)(k)

with I(d) = ∑_{l=1}^{d} h(l)l, H(d) = ∑_{m=1}^{d} h(m) and W (d) = ∑_{m=1}^{d} w_{r}(m)h(m). These two identities immediately follow from (21).

Next, we shall evaluate the function U_{r}(k_{1}, …, k_{n}), which is another representation of (16) and (18). Using the method of Tóth [20], we have the following formulas.

Theorem 6. Let k_{1}, …, k_{n} be any positive integers and let K = lcm(k_{1}, …, k_{n}) be the least common multiple of n-tuple integers k_{1}, …, k_{n}. If w_{r} is completely multiplicative function, then

U_{r}(k_{1}, …, k_{n}) = (w_{r} · t_{k_{1}}^{(1)} ⋯ t_{k_{n}}^{(n)} ∗ W_{r})(K)  (34)

with

W_{r}(d) = ∑_{{l=1 \atop (l,d)=1}}^{d} w_{r}(l),

and if w_{r} is a completely additive function, then

U_{r}(k_{1}, …, k_{n}) = (w_{r} · t_{k_{1}}^{(1)} ⋯ t_{k_{n}}^{(n)} ∗ ϕ)(K) + (t_{k_{1}}^{(1)} ⋯ t_{k_{n}}^{(n)} ∗ W_{r})(K).  (35)

Remark 7. We substitute w_{r} = 1, f_{1} = ⋯ = f_{n} = id, g_{1} = ⋯ = g_{n} = μ and h_{1} = ⋯ = h_{n} = 1 in (34) to obtain (9), and w_{r} = 1, f_{1} = ⋯ = f_{n} = f, g_{1} = ⋯ = g_{n} = g and h_{1} = ⋯ = h_{n} = 1 in (34) to obtain the formula (12). The formulas (34) and (35) are an analogue of (20) and (21), respectively.

We substitute w_{r} = id and h_{1} = ⋯ = h_{n} = 1 in (34) and use (54) below to obtain the formula (36), which is a generalization of (14). We also substitute w_{r} = log and h_{1} = ⋯ = h_{n} = 1 in (35) and use (52) below to obtain the formula (37).

Corollary 8. Let the notation be as above. Then we have

\frac{1}{K^{r}} ∑_{j=1}^{K} j^{r}s_{k_{1}}^{(1)}(j) ⋯ s_{k_{n}}^{(n)}(j)  (36)

= \frac{1}{2}s_{k_{1}}^{(1)}(K) ⋯ s_{k_{n}}^{(n)}(K) + \frac{1}{r + 1} ∑_{d|K} s_{k_{1}}^{(1)} (\frac{K}{d} ) ⋯ s_{k_{n}}^{(n)} (\frac{K}{d} ) ∑_{m=0}^{⌊r/2⌋} \binom{r + 1}{2m}B_{2m}ϕ_{1−2m}(d)

and

∑_{j=1}^{K} s_{k_{1}}^{(1)}(j) ⋯ s_{k_{n}}^{(n)}(j) log j  (37)

= (s_{k_{1}}^{(1)} ⋯ s_{k_{n}}^{(n)} · log ∗ϕ)(K) + ∑_{d|K} s_{k_{1}}^{(1)} (\frac{K}{d} ) ⋯ s_{k_{n}}^{(n)} (\frac{K}{d} ) ∑_{e|d} μ(e) Log \frac{d}{e}

− ∑_{d|K} s_{k_{1}}^{(1)} (\frac{K}{d} ) ⋯ s_{k_{n}}^{(n)} (\frac{K}{d} ) ϕ(d) ∑_{p|d} \frac{log p}{p − 1}

where Log d is given by log(d!).

As an application of Corollary 8, we provide two formulas for the weighted averages of the product gcd(k_{1}, j) gcd(k_{2}, j) ⋯ gcd(k_{n}, j) of the gcd’s.

Example 9. Let the notation be as above. Then we have

\frac{1}{K^{r}} ∑_{j=1}^{K} j^{r} gcd(k_{1}, j) ⋯ gcd(k_{n}, j)  (38)

= \frac{1}{2} gcd(k_{1}, K) ⋯ gcd(k_{n}, K)

+ \frac{1}{r + 1} ∑_{d|K} gcd (k_{1}, \frac{K}{d} ) ⋯ gcd (k_{n}, \frac{K}{d} ) ∑_{m=0}^{⌊r/2⌋} \binom{r + 1}{2m}B_{2m}ϕ_{1−2m}(d),

and

∑_{j=1}^{K} gcd(k_{1}, j) ⋯ gcd(k_{n}, j) log j  (39)

= ∑_{d|K} gcd (k_{1}, d) ⋯ gcd (k_{n}, d) (log d)ϕ (\frac{K}{d} )

+ ∑_{d|K} gcd (k_{1}, \frac{K}{d} ) ⋯ gcd (k_{n}, \frac{K}{d} ) ∑_{e|d} μ(e)Log\frac{d}{e}

− ∑_{d|K} gcd (k_{1}, \frac{K}{d} ) ⋯ gcd (k_{n}, \frac{K}{d} ) ϕ(d) ∑_{p|d} \frac{log p}{p − 1}.

We substitute w_{r} = id_{r}, f_{1} = ⋯ = f_{n} = ϕ and g_{1} = ⋯ = g_{n} = 1 in (36) to obtain (38), which is a generalization of Tóth’s result [20, Proposition 14]:

\frac{1}{K} ∑_{j=1}^{K} gcd(k_{1}, j) ⋯ gcd(k_{n}, j) = \frac{1}{K} ∑_{d|K} gcd(d, k_{1}) ⋯ gcd(d, k_{n})ϕ (\frac{K}{d} ) .

We also substitute w_{r} = log, f_{1} = ⋯ = f_{n} = ϕ and g_{1} = ⋯ = g_{n} = 1 in (37) to obtain (39).

Lastly, we shall consider some weighted averages of the product s_{k_{1}}^{(1)}(j) ⋯ s_{k_{n}}^{(n)}(j) with weights concerning the gamma function, binomial coefficients, and Bernoulli polynomials. To state Theorem 10, we use the well-known multiplication formula of Gauss–Legendre [8, Proposition 9.6.33] for the gamma function

∏_{j=1}^{n} Γ (\frac{j}{n}) = \frac{(2π)^{\frac{n−1}{2}}}{\sqrt{n}}  (40)

for any positive integer n, and the binomial formula [22, (27)]

∑_{k=0}^{[n/r]} \binom{n}{kr} = \frac{1}{r} ∑_{j=1}^{r} (1 + exp (2πi\frac{j}{r}))^{n} = \frac{2^{n}}{r} ∑_{j=1}^{r} cos^{n} \frac{jπ}{r} cos \frac{njπ}{r}  (41)

for any positive integers n and r. Furthermore, we use the well-known formula for the Bernoulli polynomial [8, Proposition 9.1.3]

∑_{j=0}^{k−1} B_{m} (\frac{j}{k} ) = \frac{B_{m}}{k^{m−1}}  (42)

for any positive integer k.

Theorem 10. Let k_{1}, …, k_{n} be any positive integers and let K = lcm(k_{1}, …, k_{n}) be the least common multiple of n-tuple integers k_{1}, …, k_{n}. Then we have

∑_{j=1}^{K} s_{k_{1}}^{(1)}(j) ⋯ s_{k_{n}}^{(n)}(j) log Γ (\frac{j}{K} )  (43)

= K log \sqrt{2π} ∑_{d_{1}|k_{1},...,d_{n}|k_{n}} \frac{f_{1}(d_{1}) ⋯ f_{n}(d_{n})}{lcm(d_{1}, …, d_{n})}g_{1} (\frac{k_{1}}{d_{1}} ) ⋯ g_{n} (\frac{k_{n}}{d_{n}} )

− (f_{1} ∗ g_{1})(k_{1}) ⋯ (f_{n} ∗ g_{n})(k_{n}) log \sqrt{2πK}

+ ∑_{d_{1}|k_{1},...,d_{n}|k_{n}} f_{1}(d_{1})g_{1} (\frac{k_{1}}{d_{1}} ) ⋯ f_{n}(d_{n})g_{n} (\frac{k_{n}}{d_{n}} ) log \sqrt{lcm(d_{1}, …, d_{n})},

∑_{j=0}^{K} \binom{K}{j}s_{k_{1}}^{(1)}(j) ⋯ s_{k_{n}}^{(n)}(j)  (44)

= 2^{K} ∑_{d_{1}|k_{1},...,d_{n}|k_{n}} \frac{f_{1}(d_{1}) ⋯ f_{n}(d_{n})}{lcm(d_{1}, …, d_{n})}g_{1} (\frac{k_{1}}{d_{1}} ) ⋯ g_{n} (\frac{k_{n}}{d_{n}} ) ×

× ∑_{l=1}^{lcm(d_{1},...,d_{n})}(−1)^{\frac{Kl}{lcm(d_{1},...,dn)}} cos^{K} \frac{lπ}{lcm(d_{1}, …, d_{n})},

and

∑_{j=0}^{K−1} B_{m} (\frac{j}{K} ) s_{k_{1}}^{(1)}(j) ⋯ s_{k_{n}}^{(n)}(j)  (45)

= \frac{B_{m}}{K^{m−1}} ∑_{d_{1}|k_{1},...,d_{n}|k_{n}} \frac{f_{1}(d_{1}) ⋯ f_{n} (d_{n})}{lcm(d_{1}, …, d_{n})^{1−m}} g_{1} (\frac{k_{1}}{d_{1}} ) ⋯ g_{n} (\frac{k_{n}}{d_{n}} ) .

As an application of Theorem 10, we give three formulas for weighted averages of the product gcd(k_{1}, j) gcd(k_{2}, j) ⋯ gcd(k_{n}, j) of the gcd’s.

Example 11. Let the notation be as above. Then we have

∑_{j=1}^{K} gcd(k_{1}, j) ⋯ gcd(k_{n}, j) log Γ (\frac{j}{K} )  (46)

= K log \sqrt{2π} ∑_{d_{1}|k_{1},...,d_{n}|k_{n}} \frac{ϕ(d_{1}) ⋯ ϕ(d_{n})}{lcm(d_{1}, …, d_{n})} + \frac{1}{2} ∑_{d_{1}|k_{1},...,d_{n}|k_{n}} ϕ(d_{1}) ⋯ ϕ(d_{n}) log lcm(d_{1}, …, d_{n})

− k_{1}k_{2} ⋯ k_{n} log \sqrt{2πK},

\frac{1}{2^{K}} ∑_{j=0}^{K} \binom{K}{j} gcd(k_{1}, j) ⋯ gcd(k_{n}, j)  (47)

= ∑_{d_{1}|k_{1},...,d_{n}|k_{n}} \frac{ϕ(d_{1}) ⋯ ϕ(d_{n})}{lcm(d_{1}, …, d_{n})} ∑_{l=1}^{lcm(d_{1},...,d_{n})}(−1)^{\frac{Kl}{lcm(d_{1},...,dn)}} cos^{K} \frac{lπ}{lcm(d_{1}, …, d_{n})}

and

∑_{j=0}^{K−1}B_{m} (\frac{j}{K} ) gcd(k_{1}, j) ⋯ gcd(k_{n}, j) = \frac{B_{m}}{K^{m−1}} ∑_{d_{1}|k_{1},...,d_{n}|k_{n}} \frac{ϕ(d_{1}) ⋯ ϕ(d_{n})}{lcm(d_{1}, …, d_{n})^{1−m}} .  (48)

We substitute f_{1} = ⋯ = f_{n} = ϕ and g_{1} = ⋯ = g_{n} = 1 in (43) and f_{1} = ⋯ = f_{n} = ϕ and g_{1} = ⋯ = g_{n} = 1 in (44) to obtain (46) and (47), respectively. We also substitute f_{1} = ⋯ = f_{n} = ϕ and g_{1} = ⋯ = g_{n} = 1 in (45) to obtain (48), which is an analogue of (30).

3 Proofs of Theorems 1, 6, 10 and Corollaries 3, 5, 8

Proof of Theorem 1. Since

t_{k_{i}}^{(i)}(j) = ∑_{d_{i}| gcd(k_{i},j)} f_{i}(d_{i})g_{i} (\frac{k_{i}}{d_{i}} ) h_{i} (\frac{j}{d_{i}} ) (i = 1, 2, …, n),

we have

U_{r}(k_{1}, …, k_{n})  (49)

= ∑_{d_{1}|k_{1},...,d_{n}|k_{n}} f_{1}(d_{1})g_{1} (\frac{k_{1}}{d_{1}} ) ⋯ f_{n}(d_{n})g_{n} (\frac{k_{n}}{d_{n}} ) ∑_{{j=1 \atop d_{1}|j,...,dn|j}}^{K} w_{r}(j)h_{1} (\frac{j}{d_{1}} ) ⋯ h_{n} (\frac{j}{d_{n}} )

= ∑_{d_{1}|k_{1},...,d_{n}|k_{n}} f_{1}(d_{1})g_{1} (\frac{k_{1}}{d_{1}} ) ⋯ f_{n}(d_{n})g_{n} (\frac{k_{n}}{d_{n}} ) ×

× ∑_{l=1}^{\frac{K}{lcm(d_{1},...,dn)}} w_{r} (l lcm(d_{1}, …, d_{n})) h_{1} (\frac{lcm(d_{1}, …, d_{n})}{d_{1}}l) ⋯ h_{n} (\frac{lcm(d_{n}, …, d_{n})}{d_{n}}l) .

We use the completely multiplicative function w_{r} in (49) to obtain

U_{r}(k_{1}, …, k_{n})  (50)

= ∑_{d_{1}|k_{1},...,d_{n}|k_{n}} f_{1}(d_{1})g_{1} (\frac{k_{1}}{d_{1}} ) ⋯ f_{n}(d_{n})g_{n} (\frac{k_{n}}{d_{n}} ) w_{r} (lcm(d_{1}, …, d_{n})) ×

× ∑_{l=1}^{\frac{K}{lcm(d_{1},...,dn)}} w_{r} (l) h_{1} (\frac{lcm(d_{1}, …, d_{n})}{d_{1}}l) ⋯ h_{n} (\frac{lcm(d_{n}, …, d_{n})}{d_{n}}l) ,

and if h_{1}, …, h_{n} are completely multiplicative functions, (50) gives

U_{r}(k_{1}, …, k_{n})

= ∑_{d_{1}|k_{1},...,d_{n}|k_{n}} f_{1}(d_{1})g_{1} (\frac{k_{1}}{d_{1}} ) h_{1} (\frac{lcm(d_{1}, …, d_{n})}{d_{1}}) ⋯ f_{n}(d_{n})g_{n} (\frac{k_{n}}{d_{n}} ) ×

× h_{n} (\frac{lcm(d_{1}, …, d_{n})}{d_{n}}) w_{r} (lcm(d_{1}, …, d_{n})) ∑_{l=1}^{\frac{K}{lcm(d_{1},...,dn)}} w_{r}(l)h_{1} (l) ⋯ h_{n} (l) .

Similarly, as in the proof of the above, using the completely additive function w_{r} in (49), we have the identity (18), and if h_{1}, …, h_{n} are completely multiplicative functions, we establish the identity (19). This completes the proof of Theorem 1.

Proof of Corollary 3. We substitute w_{r} = id_{r} and h_{1} = ⋯ = h_{n} = 1 in (17) and use

∑_{m=1}^{N} m^{r} = \frac{N^{r}}{2} + \frac{1}{r + 1} ∑_{m=0}^{⌊ \frac{r}{2} ⌋} \binom{r + 1}{2m}B_{2m}N^{r+1−2m}  (51)

for any positive integer N > 1 [8, Proposition 9.2.12], [9, Section 3.9] to obtain

∑_{j=1}^{K} j^{r}s_{k_{1}}^{(1)}(j) ⋯ s_{k_{n}}^{(n)}(j)

= ∑_{d_{1}|k_{1},...,d_{n}|k_{n}} f_{1}(d_{1})g_{1} (\frac{k_{1}}{d_{1}} ) ⋯ f_{n}(d_{n})g_{n} (\frac{k_{n}}{d_{n}} ) lcm(d_{1}, …, d_{n})^{r} ∑_{l=1}^{\frac{K}{lcm(d_{1},...,dn)}} l^{r}

= ∑_{d_{1}|k_{1},...,d_{n}|k_{n}} f_{1}(d_{1})g_{1} (\frac{k_{1}}{d_{1}} ) ⋯ f_{n}(d_{n})g_{n} (\frac{k_{n}}{d_{n}} ) lcm(d_{1}, …, d_{n})^{r}×

× \{\frac{1}{2} (\frac{K}{lcm(d_{1}, …, d_{n})})^{r} + \frac{1}{r + 1} ∑_{m=0}^{⌊ \frac{r}{2} ⌋} \binom{r + 1}{2m}B_{2m} (\frac{K}{lcm(d_{1}, …, d_{n})})^{r+1−2m}\}

= \frac{(f_{1} ∗ g_{1})(k_{1}) ⋯ (f_{n} ∗ g_{n})(k_{n})}{2}K^{r}

+ \frac{K^{r+1}}{r + 1} ∑_{m=0}^{⌊r/2⌋} \binom{r + 1}{2m}\frac{B_{2m}}{K^{2m}} ∑_{d_{1}|k_{1},...,d_{n}|k_{n}} \frac{f_{1}(d_{1}) ⋯ f_{n}(d_{n})}{lcm(d_{1}, …, d_{n})^{1−2m}} g_{1} (\frac{k_{1}}{d_{1}} ) ⋯ g_{n} (\frac{k_{n}}{d_{n}} ) ,

which proves (22). We substitute g_{1} = ⋯ = g_{n} = 1 and w_{r} = id_{r} in (16) and f_{1} = ⋯ = f_{n} = 1 and w_{r} = id_{r} in (16) to obtain the formulas for (23) and (24), respectively.

Proof of Corollary 5. We substitute w_{r} = log and h_{1} = ⋯ = h_{n} = 1 in (18) to obtain

∑_{j=1}^{K} s_{k_{1}}^{(1)}(j) ⋯ s_{k_{n}}^{(n)}(j) log j

= K ∑_{d_{1}|k_{1},...,d_{n}|k_{n}} f_{1}(d_{1})g_{1} (\frac{k_{1}}{d_{1}} ) ⋯ f_{n}(d_{n})g_{n} (\frac{k_{n}}{d_{n}} ) \frac{log lcm(d_{1}, …, d_{n})}{lcm(d_{1}, …, d_{n})}

+ ∑_{d_{1}|k_{1},...,d_{n}|k_{n}} f_{1}(d_{1})g_{1} (\frac{k_{1}}{d_{1}} ) ⋯ f_{n}(d_{n})g_{n} (\frac{k_{n}}{d_{n}} ) ∑_{l=1}^{\frac{K}{lcm(d_{1},...,dn)}} log l

= K ∑_{d_{1}|k_{1},...,d_{n}|k_{n}} f_{1}(d_{1})g_{1} (\frac{k_{1}}{d_{1}} ) ⋯ f_{n}(d_{n})g_{n} (\frac{k_{n}}{d_{n}} ) \frac{log lcm(d_{1}, …, d_{n})}{lcm(d_{1}, …, d_{n})}

+ ∑_{d_{1}|k_{1},...,d_{n}|k_{n}} f_{1}(d_{1})g_{1} (\frac{k_{1}}{d_{1}} ) ⋯ f_{n}(d_{n})g_{n} (\frac{k_{n}}{d_{n}} ) log (\frac{K}{lcm(d_{1}, …, d_{n})})!.

Furthermore, we substitute g_{1} = ⋯ = g_{n} = 1 and w_{r} = log in (18) and f_{1} = ⋯ = f_{n} = 1 and w_{r} = log in (18) to get the formulas (32) and (33), respectively.

Proof of Theorem 6. Let K = lcm(k_{1}, …, k_{n}) be the least common multiple of n-tuple positive integers k_{1}, …, k_{n}. Note that t_{k}(j) = t_{k}(gcd(k, j)) for any positive integers k and j. Since gcd(gcd(j, K), k_{i}) = gcd(j, gcd(K, k_{i})) = gcd(j, k_{i}) for any i ∈ {1, 2, …, n}, we observe that t_{k_{i}}^{(i)}(j) is equal to t_{k_{i}}^{(i)}(gcd(j, K)). If w_{r} is a completely multiplicative function, we have

U_{r}(k_{1}, …, k_{n}) = ∑_{j=1}^{K} w_{r}(j)t_{k_{1}}^{(1)}(gcd(j, K)) ⋯ t_{k_{n}}^{(n)}(gcd(j, K))

= ∑_{d|K} w_{r}(d)t_{k_{1}}^{(1)}(d) ⋯ t_{k_{n}}^{(n)}(d) ∑_{{l=1 \atop gcd(l, \frac{K}{d})=1}}^{\frac{K}{d}} w_{r}(l)

= ∑_{d|K} w_{r} (\frac{K}{d} ) t_{k_{1}}^{(1)} (\frac{K}{d} ) ⋯ t_{k_{n}}^{(n)} (\frac{K}{d} ) ∑_{{l=1 \atop gcd(l,d)=1}}^{d} w_{r}(l),

which completes the proof of the formula (34). Similarly, as in the proof of (34), we have (35).

To prove Corollary 8, we need the following formula.

Lemma 12. For any positive integers N > 1 and r, we have

∑_{{l=1 \atop gcd(l,N)=1}}^{N} log l = ∑_{d|N} μ (\frac{N}{d} ) log(d!) − ϕ(N) ∑_{p|N} \frac{log p}{p − 1}  (52)

and

∑_{m=0}^{⌊r/2⌋} \binom{r + 1}{2m}B_{2m} = \frac{r + 1}{2}.  (53)

Proof. Using the well-known identity

∑_{d|N} \frac{μ(d)}{d} log d = −\frac{ϕ(N)}{N} ∑_{p|N} \frac{log p}{p − 1}

[8, Exercise 10.8.45 (c)], we have

∑_{{l=1 \atop gcd(l,N)=1}}^{N} log l = ∑_{d|N} μ(d) ∑_{j=1}^{N/d} log (dj)

= ∑_{d|N} μ(d) log (\frac{N}{d} )! + N ∑_{d|N} \frac{μ(d)}{d} log d

= ∑_{d|N} μ(d) log (\frac{N}{d} )! − ϕ(N) ∑_{p|N} \frac{log p}{p − 1},

which completes the proof of (52). From Theorem 12.15 in [7], we have

∑_{m=0}^{r} \binom{r + 1}{m}B_{m} = 0.

It follows that

\binom{r + 1}{0}B_{0} + \binom{r + 1}{1}B_{1} + \binom{r + 1}{2}B_{2} + … + \binom{r + 1}{2[\frac{r}{2}]}B_{2[ \frac{r}{2} ]} = 0.

Since B_{0} = 1, B_{1} = −\frac{1}{2} and B_{2m+1} = 0 for any positive integer m, we obtain the formula (53).

Proof of Corollary 8. We substitute w_{r} = id_{r} and h_{1} = ⋯ = h_{n} = 1 in (34) and use the formula [16, Corollary 4]

∑_{{m=1 \atop gcd(m,N)=1}}^{N} m^{r} = \frac{N^{r+1}}{r + 1} ∑_{m=0}^{⌊ \frac{r}{2} ⌋} \binom{r + 1}{2m}B_{2m}ϕ_{1−2m}(N)  (54)

for any positive integers N > 1, and r to obtain

∑_{j=1}^{K} j^{r}s_{k_{1}}^{(1)}(j) ⋯ s_{k_{n}}^{(n)}(j)

= ∑_{d|K} d^{r}s_{k_{1}}^{(1)}(d) ⋯ s_{k_{n}}^{(n)}(d) ∑_{{l=1 \atop gcd(l, \frac{K}{d})=1}}^{K/d} l^{r}

= K^{r}s_{k_{1}}^{(1)}(K) ⋯ s_{k_{n}}^{(n)}(K) + \frac{K^{r}}{r + 1} ∑_{{d|K \atop d<K}} s_{k_{1}}^{(1)} (d) ⋯ s_{k_{n}}^{(n)} (d) ∑_{m=0}^{⌊r/2⌋} \binom{r + 1}{2m}B_{2m}ϕ_{1−2m} (\frac{K}{d} )

= K^{r}s_{k_{1}}^{(1)}(K) ⋯ s_{k_{n}}^{(n)}(K) + \frac{K^{r}}{r + 1} ∑_{{d|K \atop d>1}} s_{k_{1}}^{(1)} (\frac{K}{d} ) ⋯ s_{k_{n}}^{(n)} (\frac{K}{d} ) ∑_{m=0}^{⌊r/2⌋} \binom{r + 1}{2m}B_{2m}ϕ_{1−2m} (d)

= K^{r}s_{k_{1}}^{(1)}(K) ⋯ s_{k_{n}}^{(n)}(K) − \frac{K^{r}}{r + 1}s_{k_{1}}^{(1)}(K) ⋯ s_{k_{n}}^{(n)}(K) ∑_{m=0}^{⌊r/2⌋} \binom{r + 1}{2m}B_{2m}

+ \frac{K^{r}}{r + 1} ∑_{d|K} s_{k_{1}}^{(1)} (\frac{K}{d} ) ⋯ s_{k_{n}}^{(n)} (\frac{K}{d} ) ∑_{m=0}^{⌊r/2⌋} \binom{r + 1}{2m}B_{2m}ϕ_{1−2m} (d) .

Using the well-known identity (53), we obtain the formula (36). We also substitute w_{r} = log and h_{1} = ⋯ = h_{n} = 1 in (35), and use (52) to obtain

∑_{j=1}^{K}s_{k_{1}}^{(1)}(j) ⋯ s_{k_{n}}^{(n)}(j) log j

= (s_{k_{1}}^{(1)} ⋯ s_{k_{n}}^{(n)} · log ∗ϕ)(K) + ∑_{d|K} s_{k_{1}}^{(1)} (\frac{K}{d} ) ⋯ s_{k_{n}}^{(n)} (\frac{K}{d} ) ∑_{e|d} μ(e) log (\frac{d}{e})!

− ∑_{d|K} s_{k_{1}}^{(1)} (\frac{K}{d} ) ⋯ s_{k_{n}}^{(n)} (\frac{K}{d} ) ϕ(d) ∑_{p|d} \frac{log p}{p − 1}.

Proof of Theorem 10. Using (40) and substituting w_{r}(d) = log Γ (\frac{d}{K} ) and h_{1} = ⋯ = h_{1} = 1 in (49), we have

∑_{j=1}^{K} s_{k_{1}}^{(1)}(j) ⋯ s_{k_{n}}^{(n)}(j) log Γ (\frac{j}{K} )

= ∑_{d_{1}|k_{1},...,d_{n}|k_{n}} f_{1}(d_{1})g_{1} (\frac{k_{1}}{d_{1}} ) ⋯ f_{n}(d_{n})g_{n} (\frac{k_{n}}{d_{n}} ) ∑_{l=1}^{\frac{K}{lcm(d_{1},...,dn)}} log Γ (\frac{lcm(d_{1}, …, d_{n})}{K}l)

= ∑_{d_{1}|k_{1},...,d_{n}|k_{n}} f_{1}(d_{1})g_{1} (\frac{k_{1}}{d_{1}} ) ⋯ f_{n}(d_{n})g_{n} (\frac{k_{n}}{d_{n}} ) ×

× \{\frac{K}{lcm(d_{1}, …, d_{n})} log \sqrt{2π} − log \sqrt{2πK} + log \sqrt{lcm(d_{1}, …, d_{n})}\}

= K log \sqrt{2π} ∑_{d_{1}|k_{1},...,d_{n}|k_{n}} \frac{f_{1}(d_{1}) ⋯ f_{n}(d_{n})}{lcm(d_{1}, …, d_{n})}g_{1} (\frac{k_{1}}{d_{1}} ) ⋯ g_{n} (\frac{k_{n}}{d_{n}} )

− log \sqrt{2πK} ∑_{d_{1}|k_{1}} f_{1}(d_{1})g_{1} (\frac{k_{1}}{d_{1}} ) ⋯ ∑_{d_{n}|k_{n}} f_{n}(d_{n})g_{n} (\frac{k_{n}}{d_{n}} )

+ ∑_{d_{1}|k_{1},...,d_{n}|k_{n}} f_{1}(d_{1})g_{1} (\frac{k_{1}}{d_{1}} ) ⋯ f_{n}(d_{n})g_{n} (\frac{k_{n}}{d_{n}} ) log \sqrt{lcm(d_{1}, …, d_{n})},

which completes the proof of (43). We set w_{r}(j) = \binom{K}{j} and h_{1} = ⋯ = h_{1} = 1. Using (41), we have

∑_{j=0}^{K} \binom{K}{j}s_{k_{1}}^{(1)}(j) ⋯ s_{k_{n}}^{(n)}(j)

= ∑_{d_{1}|k_{1},...,d_{n}|k_{n}} f_{1}(d_{1})g_{1} (\frac{k_{1}}{d_{1}} ) ⋯ f_{n}(d_{n})g_{n} (\frac{k_{n}}{d_{n}} ) ∑_{m=0}^{\frac{K}{lcm(d_{1},··· ,dn)}} \binom{K}{lcm(d_{1}, …, d_{n})m}

= 2^{K} ∑_{d_{1}|k_{1},...,d_{n}|k_{n}} \frac{f_{1}(d_{1}) ⋯ f_{n}(d_{n})}{lcm(d_{1}, …, d_{n})}×

× g_{1} (\frac{k_{1}}{d_{1}} ) ⋯ g_{n} (\frac{k_{n}}{d_{n}} ) ∑_{j=1}^{lcm(d_{1},...,d_{n})} cos^{K} \frac{jπ}{lcm(d_{1}, …, d_{n})} cos \frac{Kjπ}{lcm(d_{1}, …, d_{n})}

= 2^{K} ∑_{d_{1}|k_{1},...,d_{n}|k_{n}} \frac{f_{1}(d_{1}) ⋯ f_{n}(d_{n})}{lcm(d_{1}, …, d_{n})}×

× g_{1} (\frac{k_{1}}{d_{1}} ) ⋯ g_{n} (\frac{k_{n}}{d_{n}} ) ∑_{j=1}^{lcm(d_{1},...,d_{n})}(−1)^{\frac{Kj}{lcm(d_{1},...,dn)}} cos^{K} \frac{jπ}{lcm(d_{1}, …, d_{n})},

hence, we obtain (44). Setting w_{r}(j) = B_{m} (\frac{j}{K} ) and h_{1} = ⋯ = h_{1} = 1 and using (42), we have

∑_{j=0}^{K−1} B_{m} (\frac{j}{K} ) s_{k_{1}}^{(1)}(j) ⋯ s_{k_{n}}^{(n)}(j)

= ∑_{d_{1}|k_{1},...,d_{n}|k_{n}} f_{1}(d_{1})g_{1} (\frac{k_{1}}{d_{1}} ) ⋯ f_{n}(d_{n})g_{n} (\frac{k_{n}}{d_{n}} ) ∑_{l=0}^{\frac{K}{lcm(d_{1},...,dn)} −1} B_{m} (\frac{lcm(d_{1}, …, d_{n})}{K}l)

= \frac{B_{m}}{K^{m−1}} ∑_{d_{1}|k_{1},...,d_{n}|k_{n}} \frac{f_{1}(d_{1}) ⋯ f_{n} (d_{n})}{lcm(d_{1}, …, d_{n})^{1−m}} g_{1} (\frac{k_{1}}{d_{1}} ) ⋯ g_{n} (\frac{k_{n}}{d_{n}} ) .

Hence, we prove (45).

4 Acknowledgments

The authors deeply thank the referee for carefully reading this paper and useful comments.

References

[1] E. Alkan, On the mean square averages of special values of L-functions, J. Number Theory 131 (2011), 1470–1485.

[2] E. Alkan, Distribution of averages of Ramanujan sums, Ramanujan J. 29 (2012), 385– 408.

[3] E. Alkan, Averages of values of L-series, Proc. Amer. Math. Soc 141 (2013), 1161–1175.

[4] E. Alkan, Ramanujan sums are nearly orthogonal to powers, J. Number Theory 140 (2014), 147–168.

[5] D. R. Anderson and T. M. Apostol, The evaluation of Ramanujan’s sum and generalizations, Duke Math. J. 20 (1952), 211–216.

[6] T. M. Apostol, Arithmetical properties of generalized Ramanujan sums, Pacific J. Math 41 (1972) 281–293.

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

[8] H. Cohen, Number Theory, Volume II: Analytic and Modern Tools, Graduate Texts in Mathematics, 240, Springer, 2007.

[9] L. Comtet, Advanced Combinatorics, The Art of Finite and Infinite Expansions, D. Reidel Publishing Co, 1974.

[10] A. Deitmar, S. Koyama, and N. Kurokawa, Absolute zeta functions, Proc. Japan Aca. Ser. A Math. Sci. 84 (2008), 138–142.

[11] I. Kiuchi, M. Minamide, and M. Ueda, Averages of Anderson–Apostol’s sums, J. Ramanujan Math. Soc., to appear.

[12] V. A. Liskovets, A multivariate arithmetic function of combinatorial and topological significance, Integers 10 (2010), 155–177.

[13] N. Minami, On the random variable ℕ ∋ l ↦ gcd(l, n_{1})gcd(l, n_{2}) ⋯ gcd(l, n_{k}) ∈ ℕ.

    Preprint, 2009, available at http://arxiv.org/abs/0907.0918v2.

[14] P. J. McCarthy, Introduction to Arithmetical Functions, Springer, 1986.

[15] A. D. Mednykh and R. Nedela, Enumeration of unrooted maps with given genus, J. Combin. Theory, Ser. B 96 (2006), 706–729.

[16] J. Singh, Defining power sums of n and ϕ(n) integers, Int. J. Number Theory 5 (2009), 41–53.

[17] R. Sivaramakrishnan, Classical Theory of Arithmetic Functions, Monographs and Textbooks in Pure and Applied Mathematics, Vol.126, Marcel Dekker, 1989.

[18] L. Tóth, Averages of Ramanujan sums: Note on two papers by E. Alkan, Ramanujan J. 35 (2014), 149–156.

[19] L. Tóth, Sums of products of Ramanujan sums, Annali dell’ Universita di Ferrara, 58 (2012), 183–197.

[20] L. Tóth, Some remarks on a paper of V. A. Liskovets, Integers: Elec. J. Combinatorial Number Theory 11 (2011), Paper #A51.

[21] L. Tóth, Menon’s identity and arithmetical sums representing functions of several variables, Rend. Semin. Mat. Univ. Politec. Torino 69 (2011), 97–110.

[22] L. Tóth, Weighted gcd-sum functions, J. Integer Sequences 14 (2011), Article 11.7.7.

2010 Mathematics Subject Classification: Primary 11A25; Secondary 11B68. Keywords: arithmetical function, Ramanujan’s sum, greatest common divisor.

(Concerned with sequences \underline{A018804}, \underline{A051193}, \underline{A056188}, and \underline{A159068}.)

Received August 18 2015; revised versions received November 21 2015; March 17 2016. Published in Journal of Integer Sequences, March 18 2016.

Return to Journal of Integer Sequences home page.