On Certain Sums of Arithmetic Functions Involving the GCD and LCM of Two Positive Integers

We obtain asymptotic formulas with remainder terms for the hyperbolic summations ∑mn≤xf((m,n))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{mn\le x} f((m,n))$$\end{document} and ∑mn≤xf([m,n])\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{mn\le x} f([m,n])$$\end{document}, where f belongs to certain classes of arithmetic functions, (m, n) and [m, n] denoting the gcd and lcm of the integers m, n. In particular, we investigate the functions f(n)=τ(n),logn,ω(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(n)=\tau (n), \log n, \omega (n)$$\end{document} and Ω(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega (n)$$\end{document}. We also define a common generalization of the latter three functions, and prove a corresponding result.


Introduction
Let F : N 2 → C be an arithmetic function of two variables. Several asymptotic results for sums F (m, n) with various bounds of summation are given in the literature. The usual 'rectangular' summations are of form m≤x, n≤y F (m, n), in particular with x = y. The 'triangular' summations can be written as The 'hyperbolic' summations have the shape mn≤x F (m, n), the sum being over the Dirichlet region {(m, n) ∈ N 2 : mn ≤ x}. Hyperbolic summations have been less studied than rectangular and triangular summations and it is hyperbolic summations that are estimated in this paper.
We mention a few examples for functions F involving the greatest common divisor (gcd) and the least common multiple (lcm) of integers. If F (m, n) = (m, n), the gcd of m and n, then holds for every ε > 0, where ζ is the Riemann zeta function, ζ is its derivative, γ is Euler's constant, and θ denotes the exponent appearing in Dirichlet's divisor problem. Furthermore, mn≤x (m, n) = 1 4ζ (2) x(log x) 2 where c 1   For other related asymptotic results for functions involving the gcd and lcm of two (and several) integers see the papers [2,8,11,17,19] and their references. For summations of some other functions of two variables see [3,14,20].
We remark that for any arithmetic function f : n)). Hence to find estimates for (1.5) is, in fact, a one variable summation problem. Formula (1.2) represents its special case when f (k) = k (k ∈ N). The sum of divisors function f (k) = σ(k), giving an estimate similar to (1.2), with the same error term, was considered in paper  [11]. The case of the divisor function f (k) = τ (k) was discussed by Heyman [7], obtaining an asymptotic formula with error term O(x 1/2 ) by using elementary estimates. However, for every k ∈ N, mn=k τ ((m, n)) =  [12] using deep analytic methods. We deduce the following result.
In an analogous manner to (1.5), let us define We remark that if F is an arbitrary arithmetic function of two variables, then the one variable function is called the convolute of F . The function F of two variables is said to be multiplicative if F (m 1 m 2 , n 1 n 2 ) = F (m 1 , n 1 )F (m 2 , n 2 ) provided that (m 1 n 1 , m 2 n 2 ) = 1. If F is multiplicative, then F is also multiplicative. See Vaidyanathaswamy [21], Tóth [18,Sect. 6]. The functions G f and L f of above are special cases of this general concept. If f is multiplicative, then G f (k) and L f (k) are multiplicative as well.
In the present paper we deduce simple arithmetic representations of the functions G f (k) and L f (k) (Proposition 2.1), and establish new asymptotic estimates for sums of type (1.5) and (1.7). Namely, we give estimates for mn≤x f ((m, n)) when f belongs to a wide class of functions (Theorem 2.2), and obtain better error terms in the case of a narrower class of functions (Theorem 2.4). In particular, we consider the functions f (n) = log n, ω(n) and Ω(n) (Corollary 2.6). Actually, we define a common generalization of these three functions and prove a corresponding result (Corollary 2.5). We also point out the case of the function f (n) = 1/n, the related result on mn≤x (m, n) −1 (Corollary 2.3) being strongly connected with the sum

Useful arithmetic representations of the functions
, already defined in the Introduction, are given by the next result.

3)
and If f is additive, then for every n ∈ N, If f is completely additive, then for every n ∈ N, See [11,Prop. 5.1] for a similar formula on the sum d1···d k =n g((d 1 , . . . , d k )), where k ∈ N and g is an arithmetic function.
Our first asymptotic formula applies to every function f satisfying a condition on its order of magnitude.
where the constants C f and D f are given by with C defined by , (2.9) and the error term is The error term R f (x) can be improved assuming that the Riemann Hypothesis (RH) is true. For example, let = 221/608 Theorem 2.2 applies, e.g., to the functions f (n) = n β (with β < 1, or Ω(n) (with 0 < β = ε arbitrary small, δ = 0). We point out the case of the function f (n) = n −1 .
where is given in Theorem 2.2.
However, for some special functions asymptotic formulas with more terms or with better unconditional errors can be obtained. See, e.g. (1.6), namely the case f (n) = τ (n) and our next results.
Let f be a function such that (μ * f )(n) = 0 for all n = p ν (n is not a prime power), (μ * f )(p ν ) = g(p) does not depend on ν and g(p) is sufficiently small for the primes p. More exactly, we have the next result.

Theorem 2.4. Let f be an arithmetic function such that there exists a subset
Q of the set of primes P and there exists a subset S of N with 1 ∈ S, satisfying the following properties: , depending only on p, for all prime powers p ν with p ∈ Q, ν ∈ S. iii) g(p) (log p) η , as p → ∞, where η ≥ 0 is a fixed real number. Then for the error term in (2.8) we have R f (x) x 1/2 (log x) η . Furthermore, the constants C f and D f can be given as The prototype of functions f to which Theorem 2.4 applies is the function f S,η implicitly defined by where 1 ∈ S ⊆ N, η ≥ 0 is real and Q = P. It is possible to consider the corresponding generalization with Q ⊂ P, as well. By Möbius inversion we obtain that for n = p p νp(n) ∈ N, According to (2.12), the conditions of Theorem 2.4 are satisfied and we deduce the next result.

Corollary 2.5. If 1 ∈ S ⊆ N and η ≥ 0 is a real number, then
where the constants C fS,η and D fS,η are given by In the special cases mentioned above we obtain the following results.
Now consider the functions given by L f (n) = ab=n f ([a, b]).

If RH is true, then the error term is O(x 1+ +ε ), where is given in Theorem 2.2.
If the function f is (completely) additive, then identities (2.6) and (2.7) can be used to deduce asymptotic estimates for n≤x L f (n).
where C log and D log are given by (2.17).
As a consequence we deduce that mn≤x [m, n] ∼ x x log x as x → ∞.  (2.20) where C ω is given by (2.18) and where C Ω is given by (2.19) and Next we consider the divisor function f (n) = τ (n).
for every ε > 0, where = 0.078613, and the constants C 2 , C 3 , C 4 can also be given explicitly.
Finally, we deduce the counterpart of formula (1.4) with hyperbolic summation.

Proofs
Proof of Proposition 2.1. Group the terms of the sum G f (n) = ab=n f ((a, b)) according to the values (a, b) = d, where a = dc, b = de with (c, e) = 1. We obtain, using the property of the Möbius μ function, giving (2.1), which can be written as (2.2) and (2.3) by the definition of the Dirichlet convolution and the identity δ 2 t=k μ(δ)τ (t) = 2 ω(k) . Alternatively, use the identity giving (2.2). For L f (n) use that [a, b] = ab/(a, b) and apply the first method above to deduce (2.4) and (2.5).
If f is an additive function, then f ((a, b))+f ([a, b]) = f (a)+f (b) holds for every a, b ∈ N. To see this, it is enough to consider the case when a = p r , b = p s are powers of the same prime p. Now, f (p min(r,s) )+f (p max(r,s) ) = f (p r )+f (p s ) trivially holds for every r, s ≥ 0. Therefore, which is (2.6). For the proof of Theorem 2.2 we need the following Lemmas.
Proof. The function t → t −s (log t) δ (t > x) is decreasing for large x. By comparing the sum with the corresponding integral we have If δ < 0, then trivially, If δ > 0, then integrating by parts gives and repeated applications of the latter estimate, until the exponent of log t becomes negative, conclude the proof.