Sums of averages of gcd-sum functions II

Let $\gcd(k,j)$ denote the greatest common divisor of the integers $k$ and $j$, and let $r$ be any fixed positive integer. Define $$ M_r(x; f) := \sum_{k\leq x}\frac{1}{k^{r+1}}\sum_{j=1}^{k}j^{r}f(\gcd(j,k)) $$ for any large real number $x\geq 5$, where $f$ is any arithmetical function. Let $\phi$, and $\psi$ denote the Euler totient and the Dedekind function, respectively. In this paper, we refine asymptotic expansions of $M_r(x; {\rm id})$, $M_r(x;{\phi})$ and $M_r(x;{\psi})$. Furthermore, under the Riemann Hypothesis and the simplicity of zeros of the Riemann zeta-function, we establish the asymptotic formula of $M_r(x;{\rm id})$ for any large positive number $x>5$ satisfying $x=[x]+\frac{1}{2}$.


Introduction and Statement of Results
Let gcd(k, j) be the greatest common divisor of the integers k and j. The gcd-sum function, which is also known as Pillai's arithmetical function, is defined by P (n) = n k=1 gcd(k, n).
This function has been studied by many authors such as Broughan [3], Bordellés [4],Tanigawa and Zhai [18], Tóth [19], and others. Analytic properties for partial sums of the gcd-sum function f (gcd(j, k)) were recently studied by Inoue and Kiuchi [7,8]. We recall that the symbol * denotes the Dirichlet convolution of two arithmetical functions f and g defined by f * g(n) = d|n f (d)g(n/d), for every positive integer n. For any arithmetical function f , the second author [11] showed, that for any fixed positive integer r and any large positive number x ≥ 2 we have Here, as usual, the function µ denotes the Möbius function and B m = B m (0) are the Bernoulli numbers, with B m (x) being the Bernoulli polynomials defined by the generating function ze xz with |z| < 2π. Many applications of Eq. (1.1) have been given in [10], [12] and [13].
The first purpose of this paper is to refine the error terms K r (x), L r (x) and U r (x) from the above formulas. Therefore, let σ u = id u * 1 be the generalized divisor function for any real number u and let m ≥ 1 be an integer. Then for any large positive number x ≥ 2, the function ∆ −2m (x) denotes the error term of the generalized divisor problem given by We have the following results: where the function δ(x) is defined by with C being a positive constant.
Moreover, we have Furthermore, even better estimates of K r (x) can be achieved by additional assumptions on the Riemann zeta-function. Under the Riemann Hypothesis, a sharper estimate of the partial sum of the Möbius function has been given by Soundararajan [17], who proved that for any large positive number x > 5 satisfying x = [x] + 1 2 . This result has later been improved by Maier and Montgomery [15] and by Balazard and de Roton [2]. By using the above result on M(x), we obtain the next statement.
For our further considerations, let ρ = β + iγ denote the generic non-trivial zeros of the Riemann zeta-function. Under the assumption that all zeros ρ in the critical strip of ζ(s) are simple, we are able to prove an additional refinement for the error term K r (x). Theorem 1.3. Assume that the zeros of ζ(s) are simple. Let T * ≥ x 6 be some positive number satisfying the inequality where the functions C odd (r) and C even (r) are given by for any fixed positive integer r.
Finally, define the sum which is intimately connected to Mertens function. Assuming the simplicity of the zeros of ζ(s), Gonek [5] and Hejhal [6] independently conjectured that for any real number λ < 3/2, we have We use this conjecture to prove the following: Theorem 1.4. Assume that the Riemann Hypothesis and Gonek-Hejhal conjecture. Then for any large positive number x > 5 satisfying x = [x] + 1 2 .

Proofs of Theorems 1.1 and 1.2
In order to prove our main results, we first show some necessary lemmas.
This completes the proof.
Lemma 2.5. For any large positive number x > 5, we have Now, we write our sums as follows To complete the proof, it remains to estimate the last sum above. Notice that and that and Eq. (2.12) is proved.
for any positive integer m.
Proof. We use the identity x log x + 1 which completes the proof of Eq. (2.17). By using the fact that together with Eqs. (1.6), (2.7), and (2.9) we get for any positive integer m.
Proof. From the identity µ * ψ id