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≤x f((m,n)) and ∑ mn≤x f([m,n]), 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), log n, ω(n) and Ω(n). We also define a common generalization of the latter three functions, and prove a corresponding result. 2010 Mathematics Subject Classification: 11A05, 11A25, 11N37


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 n≤x m≤n F (m, n). Note that if the function F is symmetric in the variables, then 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.
If F (m, n) = [m, n], the lcm of m and n, then we have For other related asymptotic results for functions involving the gcd and lcm of two (and several) integers see the papers [2,8,11,15,17] and their references. For summations of some other functions of two variables see [3,13,18].
We remark that for any arithmetic function f : where G f (k) = mn=k f ((m, 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)) = where the best known related error term, to our knowledge, is R(x) ≪ x 63/178+ε , with 63/178 . = 0.353932, given by Liu [12] using deep analytic methods.
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 [19], Tóth [16,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 2.3). In particular, we consider the functions f (n) = log n, ω(n) and Ω(n) (Corollary 2.5). Actually, we define a common generalization of these three functions and prove a corresponding result (Corollary 2.4). We also investigate the function f (n) = 1/n, the related result on mn≤x (m, n) −1 (Theorem 2.6) being strongly connected with the sum mn≤x [m, n] (Theorem 2.7). Furthermore, we deduce estimates for the sums mn≤x f ([m, n]) in the cases of f (n) = log n, Ω(n) (Theorems 2.8, 2.9) and f (n) = τ (n) (Theorem 2.10), respectively. We pose as an open problem to obtain an estimate for the sum mn≤x ω([m, n]). Finally we obtain a formula for mn≤x (m, n)[m, n] −1 (Theorem 2.11). The proofs are given in Section 3.

Main results
Useful arithmetic representations of the functions G f (n) = ab=n f ((a, b)) and L f (n) = ab=n f ([a, b]), already defined in the Introduction, are given by the next result.
Proposition 2.1. Let f be an arbitrary arithmetic function. Then for every n ∈ N,

3)
and If f is completely additive, then for every n ∈ N, In terms of formal Dirichlet series, identities (2.1), (2.2) and (2.3) show that for every arithmetic function f , See [11, Prop. 5.1] for a similar formula on the sum d 1 ···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 8) and the error term is The error term R f (x) can be improved assuming that the Riemann Hypothesis is true. For example, let ̺ = 221/608 or Ω(n) (with 0 < β = ε arbitrary small, δ = 0). 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 theorems.
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 2.3. 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: Then for the error term in (2.7) 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 2.3 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, where f S,η (1) = 0 (empty sum).
where the constants C f S,η and D f S,η are given by In the special cases mentioned above we obtain the following results. (2.14) We deduce by (2.10) and (2.11) that mn≤x (m, n) ∼ x C log x and mn≤x κ((m, n)) ∼ x C log κ x , as x → ∞.
The next result concerns the function f (n) = 1/n and gives a better error term than the error obtained from Theorem 2.2.
and θ is the exponent in Dirichlet's divisor problem.
and θ is the exponent in Dirichlet's divisor problem.
If the function f is completely additive, then identity (2.6) can be used to deduce asymptotic estimates for n≤x L f (n).
where C log and D log are given by (2.13).
As a consequence we deduce that mn≤x [m, n] ∼ x x log x as x → ∞.
where C Ω is given by (2.14) and Γ being the Gamma function.
Our treatment cannot be applied to the function f (n) = ω(n), which is additive, but not completely additive. We formulate as an open problem to deduce an estimate for mn≤x ω([m, n]).
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.

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 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 f (n) = d|n (f * µ)(d) to deduce that giving (2.2). /(a, b) and apply the first method above to deduce (2.4) and (2.5).
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.
Proof. Let 0 < s < 1. If δ ≥ 0, then trivially n≤x (log n) δ n s ≤ (log x) δ n≤x 1 n s and by comparison of the sum with the corresponding integral we have which is, using again (3.1), where 1 − s > 0. The case s = 1 is well-known. If s > 1, then the corresponding series is convergent.
Proof of Theorem 2.2. We use identity (2.3) and the known estimate (see [6]) where C is defined by (2.8). We deduce by standard arguments that mn≤x f ((m, n)) = Here where the series converges absolutely by the given assumption on f , and where the series converges absolutely and by using Lemma 3.1 again, giving the same error x (β+1)/2 (log x) δ+1 . Finally, using Lemma 3.2.
Baker [1] proved that under the Riemann Hypothesis for the error term R(x) of estimate (3.2) one has R(x) ≪ x 4 11 +ε , whilst Kaczorowski and Wiertelak [9] remarked that a slight modification of the treatment in [20] yields R(x) ≪ x 221 608 +ε . This leads to the desired improvement of the error. Now we will prove Theorem 2.3. We need the following Lemmas. Proof. We have, by using Riemann-Stieltjes integration, integration by parts and the Chebyshev estimate π(x) ≪ x/ log x, Integration by parts, again, gives and repeated applications of the latter estimate conclude the result.
using that 1 ∈ S. Here where the series is absolutely convergent by the condition g(p) ≪ (log p) η and by (3.5), and the last sum is by Lemma 3.3. Also, .