Sums of Averages of GCD-Sum Functions II

Let gcd(k,j)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \gcd (k,j) $$\end{document} denote the greatest common divisor of the integers k and j, and let r be any fixed positive integer. Define Mr(x;f):=∑k≤x1kr+1∑j=1kjrf(gcd(j,k))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} M_r(x; f) := \sum _{k\le x}\frac{1}{k^{r+1}}\sum _{j=1}^{k}j^{r}f(\gcd (j,k)) \end{aligned}$$\end{document}for any large real number x≥5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x\ge 5$$\end{document}, where f is any arithmetical function. Let ϕ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi $$\end{document}, and ψ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi $$\end{document} denote the Euler totient and the Dedekind function, respectively. In this paper, we refine asymptotic expansions of Mr(x;id)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_r(x; \mathrm{id})$$\end{document}, Mr(x;ϕ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_r(x;{\phi })$$\end{document} and Mr(x;ψ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_r(x;{\psi })$$\end{document}. Furthermore, under the Riemann Hypothesis and the simplicity of zeros of the Riemann zeta-function, we establish the asymptotic formula of Mr(x;id)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_r(x;\mathrm{id})$$\end{document} for any large positive number x>5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x>5$$\end{document} satisfying x=[x]+12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x=[x]+\frac{1}{2}$$\end{document}.


Introduction and Statement of Results
Let gcd(k, j) be the greatest common divisor of the integers k and j. The gcdsum function, which is also known as Pillai's arithmetical function, is defined 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 with |z| < 2π. Many applications of Eq. (1) have been given in [10], [12] and [13].
In [11], Eq. (1) was used to establish asymptotic formulas for M r (x; f ) for specific choices of f such as the identity function id, the Euler totient function φ = id * μ or the Dedekind function ψ = id * |μ|. More precisely, let ζ(s) denote the Riemann zeta-function, then for f = id, it was proved that M r (x; id) = 1 (r + 1)ζ (2) x log x + x 2 For f = φ, it was shown that M r (x; φ) = 1 (r + 1)ζ 2 (2) x log x + x 2ζ(2) Lastly, for f = ψ it was proved that The function Δ(x) denotes the error term of the Dirichlet divisor problem: Let τ = 1 * 1 be the divisor function, then for any large positive number x ≥ 2, where γ is the Euler constant and Δ(x) can be estimated by Δ(x) = O x θ+ε . It is known that one can take 1/4 ≤ θ ≤ 1/3. More precisely, the Dirichlet divisor problem is to find the smallest value of θ for which the above estimate holds, for any > 0. This problem is still unsolved. The best estimate to date is O x 131/416 (log x) 26947/8320 , obtained by Huxley [7] in 2003. 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 Δ −2m (x) denotes the error term of the generalized divisor problem given by For more details about the functions Δ(x), Δ −2m (x), see [1]. We have the following results: be the error terms given by Eqs. (5) and (6), respectively. For any large positive number x > 5 and fixed positive integer r, we have where the function δ(x) is defined by with C being a positive constant. Moreover, we have

Remark 1.
It is easily checked that using the weakest estimate Δ −2m (x) = O m (1) in the results Theorem 1 yields much better results than the previously known formulas for K r (x), L r (x) and M r (x) from Eqs. (2), (3), and (4).
Furthermore, even better estimates of K r (x) can be achieved by additional assumptions on the Riemann zeta-function. Under the Riemann Hypothesis, Maier and Montgomery [15] gave a sharper estimate of the partial sum of the Möbius function, which was later improved by Soundararajan [17]. The author proved that for any large positive number x > 5 satisfying x = [x]+ 1 2 . This latter has been improved slightly by Balazard and de Roton [2]. By using the above result on M (x), we obtain the next statement.
Theorem 2. Assume the Riemann Hypothesis and let Δ(x) and Δ −2m (x) be the error terms given by Eqs. (5) and (6), respectively. Then for any large positive number x > 5 such that x = [x] + 1 2 and fixed positive integer r, we have For our further considerations, let ρ = α + iβ denote the generic nontrivial 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 3.
Assume that the zeros of ζ(s) are simple. Let T * ≥ x 6 be some positive number satisfying the inequality 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:

Proofs of Theorems 1 and 2
In order to prove our main results, we first show some necessary lemmas.

Auxiliary Lemmas Lemma 1. For any large positive number
where δ(x) is given by Eq. (7). Assume that x = [x] + 1 2 . Under the Riemann Hypothesis we have and for any large positive number x > 5. Here η(x) is given by Eq. (8).

Lemma 3. For any large positive number x > 5, we have
Proof. The proof can be found in [20,Satz 3].

Lemma 4. For any large positive number x > 5, we have
and where κ(x) is given by with D being a positive constant.

Lemma 7.
For any large positive number x > 5, we have n≤x d|n and d ≤x μ * φ(d) d for any positive integer m. Here κ(x) is defined above in Lemma 4.