1 Introduction and main results

Let $\gcd (n_{1},\ldots ,n_{k})$ be the greatest common divisor of positive integers $n_{1},\ldots ,n_{k}$ for any fixed integer $k\geq 2$ . In 2012, Krätzel et al. [Reference Krätzel, Nowak and Tóth8] considered the asymptotic behavior of the summatory function

(1.1) $$ \begin{align} S_{k}(x):= \sum_{n_{1}\ldots n_{k}\leq x} \gcd(n_{1},\ldots,n_{k}) \end{align} $$

for any real number $x>1$ and any fixed integer $k\geq 2$ . The sum runs over the hyperbolic region $ \{ (n_{1},\ldots ,n_{k}) \in {\mathbb N}^{k}:\ n_{1}\ldots n_{k}\leq x,\ k\geq 2 \}. $ Let $s=\sigma +it$ be a complex variable, and let $\zeta (s)$ denote the Riemann zeta function. For $k=2$ , they showed that

$$ \begin{align*} S_{2}(x) &= \underset{s=1}{\mathrm{Res}}~\frac{\zeta^{2}(s)\zeta(2s-1)}{\zeta(2s)}\frac{x^{s}}{s} + E_{2}(x), \end{align*} $$

where $E_{2}(x)$ is the error term, which is estimated by $O\left (x^{\theta }(\log x)^{\theta '}\right )$ for some real numbers $\frac 12 < \theta <1$ and $\theta '>0$ . Using the discrete Hardy–Littlewood method [Reference Huxley4], they obtained the values: $\theta =\frac {547}{832}(=\frac 12+\frac {131}{832})$ and $\theta '=\frac {26947}{8320}$ (see also [Reference Tóth14, Reference Tóth and Zhai15]). For any fixed integer M, Kühleitner and Nowak [Reference Kühleitner and Nowak9] studied the summation of the arithmetic functions $h_{M}(n)$ associated with the Dirichlet series

(1.2) $$ \begin{align} \sum_{n=1}^{\infty}\frac{h_{M}(n)}{n^s} &= \zeta^{2}(s)\zeta(2s-1)\zeta^{M}(2s) \end{align} $$

for $\mathrm {Re}~s>1$ . Let $G_{M}(x)$ denote the error term defined by

(1.3) $$ \begin{align} \sum_{n\leq x}h_{M}(n) &= \underset{s=1}{\mathrm{Res}}~{\zeta^{2}(s)\zeta(2s-1)\zeta^{M}(2s)}\frac{x^{s}}{s} + G_{M}(x) \end{align} $$

for any real number $x>1$ . They considered a lower bound for the error term $G_{M}(x)$ and proved the Omega-estimate for the error term:

(1.4) $$ \begin{align} G_{M}(x) =\Omega\left(x^{\frac12}\frac{(\log x)^{2}}{(\log\log x)^{|M+1|}}\right) \quad \mathrm{as} \ \ x\to \infty, \end{align} $$

which is a precise estimate and very sharp result.

In this study, we, inspired by their work, first investigate the mean square estimate of $G_{M}(x)$ . Set the integral

$$ \begin{align*}J_{M}(T) := \int_{1}^{T}G_{M}^{2}(u)du \end{align*} $$

for any real number $T\geq 16$ . By (1.4), we have

$$\begin{align*}J_{M}(T)=\Omega\left(T^{2}\frac{(\log T)^{4}}{(\log\log T)^{2|M+1|}}\right) \quad \mathrm{as} \ \ T\to \infty.\end{align*}$$

To evaluate this, we employ the properties of the Riemann zeta function and Parseval’s identity (see (A.5) in [Reference Ivić5]) for Mellin transforms. This approach, which originates from the work of Titchmarsh, has since been utilized by many authors (e.g., Hilberdink [Reference Hilberdink3]). Using this method, we derive the following result.

Theorem 1.1 Let the notation be as above. For any real number $T\geq 16$ and any fixed integer M, we have

(1.5) $$ \begin{align} J_{M}(T) & \ll_{} T^2(\log T)^{7+2|M|}. \end{align} $$

Let $\mu $ denote the Möbius function, defined by

$$ \begin{align*} \mu(n)=\left\{\begin{array}{@{}cl} \displaystyle 1, &\ \ \ \mathrm{if} \ n=1, \ \ \\ \displaystyle (-1)^k ,&\ \ \ \mathrm{if} \ n\ \mathrm{is\ squarefree\ and}\ n=p_1p_2\ldots p_k, \ \ \\ \displaystyle 0, &\ \ \ \mathrm{if} \ n\ \mathrm{is\ not\ squarefree}. \\ \end{array} \right. \end{align*} $$

Let $\sigma $ and $\phi $ define the sum-of-divisors function and the Euler totient function, defined by the Dirichlet convolutions $\mathbf {1}*\mathrm {id}$ and ${\mu }*\mathrm {id}$ , respectively, where $\mathrm {id}(n)=n$ and $\mathbf {1}(n)=1$ for any positive integer n. From Proposition 5.1 in [Reference Krätzel, Nowak and Tóth8], we have

$$ \begin{align*}h_{0}(n)=\sum_{ab=n}\sigma(\gcd(a,b)) \quad \mathrm{and} \quad h_{-2}(n)=\sum_{ab=n}\phi(\gcd(a,b)) \end{align*} $$

(see also [Reference Kühleitner and Nowak9]). Using equation (1.3) and the previous equations, the integrals $J_{0}(T)$ and $J_{-2}(T)$ represent the mean square estimates for the error terms in the partial sums of the functions $\sigma $ and $\phi $ , respectively, involving the greatest common divisor. Thus, we obtain the bounds

$$\begin{align*}J_{0}(T) \ll T^2(\log T)^{7} \ \ \mathrm{and } \ \ J_{-2}(T) \ll T^2(\log T)^{11}. \end{align*}$$

Using the Cauchy–Schwarz inequality and (1.5), we deduce the following.

Remark 1.2 For any real number $T\geq 16$ , and any fixed integer M, we have

(1.6) $$ \begin{align} \frac{1}{T}\int_{1}^{T}G_{M}(u)du & \ll_{} T^{\frac12}(\log T)^{\frac{7}{2}+|M|}. \end{align} $$

Subsequently, we may conjecture that the order of magnitude of $G_{M}(x)$ is given by the estimation on the right-hand side of (1.6).

Remark 1.3 If $J_{M}(T)$ can be evaluated using

$$ \begin{align*}c(M)T^{2}(\log T)^{\ell} + O(T^{2}(\log T)^{\ell-\delta}) \end{align*} $$

as $T \to \infty $ with a constant $c(M)>0$ , depending on M and numbers $\ell>0$ , $\delta>0$ , then we obtain the following Omega-result of $G_{M}(x)$ :

(1.7) $$ \begin{align} G_{M}(x) = \Omega\left(x^{\frac12}(\log x)^{\frac{\ell}{2}}\right). \end{align} $$

Hence, an interesting relation is revealed between equations (1.4) and (1.7). It can be observed that (1.5) provides a sharp estimate, though it is not the best one.

Moreover, by using the relationships in (1.1), (2.2), and (2.3), along with the Cauchy residue theorem to obtain

$$ \begin{align*} S_{k}(x) &= M_{k}(x) + \frac{1}{2\pi i}\int_{\alpha_{k}-i\infty}^{\alpha_{k}+i\infty} \frac{\zeta^{k}(s)\zeta(ks-1)}{\zeta(ks)s}\left(\frac{1}{x}\right)^{-s}ds, \end{align*} $$

where $\alpha _{k}$ is defined by (5.1) below, and $M_{k}(x)$ is the main term, which is given by

$$ \begin{align*} M_{k}(x)=\left\{\begin{array}{@{}cl} \displaystyle \sum_{s_{0}=1, 2/k}\underset{s=s_{0}}{\mathrm{Res}}~\frac{\zeta^{k}(s)\zeta(ks-1)}{\zeta(ks)}\frac{x^{s}}{s} &\ \ \ \mathrm{if} \ k=3,4, \\ \displaystyle \underset{s=1}{\mathrm{Res}}~\frac{\zeta^{k}(s)\zeta(ks-1)}{\zeta(ks)}\frac{x^{s}}{s} &\ \ \ \mathrm{if} \ k=5,6,7,\ldots \end{array} \right. \end{align*} $$

Then, the function $E_{k}(x)$ provides the error term of $S_{k}(x)$ , as

(1.8) $$ \begin{align} E_{k}(x) & = \frac{1}{2\pi i}\lim_{T\to \infty}\int_{\alpha_{k}-iT}^{\alpha_{k}+iT} \frac{\zeta^{k}(s)\zeta(ks-1)}{\zeta(ks)s}\left(\frac{1}{x}\right)^{-s}ds \end{align} $$

for $\alpha _{k} <1$ and is close to $1$ . For $k=3$ , Krätzel et al. [Reference Krätzel, Nowak and Tóth8] derived the order of magnitude of the error term $E_{3}(x)$ :

(1.9) $$ \begin{align} E_{3}(x) = O\left(x^{\frac12}(\log x)^{5}\right). \end{align} $$

Assume that the Riemann hypothesis (RH) is true. The second goal of this article is to consider the estimate for the mean square of $E_{k}(x)$ under the RH, which is given by (1.8), for any fixed integer $k\geq 3$ . We shall derive the integral of $I_{k}(T)$ , defined as

(1.10) $$ \begin{align} I_{k}(T) &:=\int_{1}^{T}E_{k}^{2}(u)du \end{align} $$

for any fixed integer $k\geq 3$ and any real number $T\geq 16$ . We use the Parseval identity for Mellin transforms and some properties of the Riemann zeta function under the RH to obtain the following.

Theorem 1.4 Assume that the RH is true. For any real number $T\geq 16$ and any fixed integer $k~(3 \leq k < \frac {1}{6C}\log \log T)$ with a constant $C\ (0<C<\frac {1}{18})$ , we have

(1.11) $$ \begin{align} I_{k}(T) &\ll \left\{\begin{array}{@{}cl} \displaystyle T^{\frac{11}{6}} \mathrm{exp}\left(A\frac{\log T}{\log\log T}\right), & \ \mathrm{for} \ k=3, \\ \displaystyle T^{2 - \frac{1}{k}} \mathrm{exp}\left(A'\frac{\log T}{\log\log T}\right), & \ \mathrm{for} \ k=4,5,6,7,\ldots, \end{array} \right. \end{align} $$

for some positive numbers A and $A'$ .

If $I_{k}(T)$ may be evaluated by

$$ \begin{align*} I_{k}(T) & = \left\{\begin{array}{@{}cl} \displaystyle c'T^{\frac{11}{6}}(\log T)^{l'} + O(T^{\frac{11}{6}}(\log T)^{l'-\delta'}), & \ \mathrm{for} \ k=3, \\ \displaystyle c"T^{2- \frac{1}{k}}(\log T)^{l"} + O(T^{2- \frac{1}{k}}(\log T)^{l"-\delta"}), & \ \mathrm{for} \ k=4,5,6,7,\ldots, \end{array} \right. \end{align*} $$

as $x\to \infty $ with constants $c', c">0$ and numbers $l', l">0$ , and $\delta ', \delta ">0$ , then we can deduce the Omega-result for $E_{k}(x)$ as

(1.12) $$ \begin{align} E_{k}(x) & = \left\{\begin{array}{@{}cl} \displaystyle \Omega\left(x^{\frac{5}{12}}(\log x)^{\frac{l'}{2}}\right), & \ \mathrm{for} \ k=3, \\ \displaystyle \Omega\left(x^{\frac12- \frac{1}{2k}}(\log x)^{\frac{l"}{2}}\right), & \ \mathrm{for} \ k=4,5,6,7,\ldots. \end{array} \right. \end{align} $$

The Omega-result (1.12) for $k\geq 4$ is completely analogous to the conjecture (2.1) stated below for the Piltz divisor problem (see [Reference Hilberdink3, Reference Ivić5, Reference Takeda10, Reference Takeda11, Reference Titchmarsh13]).

Using Cauchy–Schwarz’s inequality and (1.11), we deduce the following.

Remark 1.5 Let the notations be defined as above. For any real number $T\geq 16$ , we have

(1.13) $$ \begin{align} \qquad\frac{1}{T}\int_{1}^{T}E_{k}(u)du &\ll \left\{\begin{array}{@{}cl} \displaystyle T^{\frac{5}{12}}\mathrm{exp}\left(\frac{A}{2}\frac{\log T}{\log\log T}\right) & \ \mathrm{for} \ k=3, \\ \displaystyle T^{\frac12 -\frac{1}{2k}} \mathrm{exp}\left(\frac{A'}{2}\frac{\log T}{\log\log T}\right) & \ \mathrm{for} \ k=4,5,6,7,\ldots \end{array} \right. \end{align} $$

Then, we may conjecture that the order of magnitude of $E_{k}(x)$ is given by the estimates on the right-hand side of (1.13) under the RH, namely,

(1.14) $$ \begin{align} {E}_{k}(x) &\ll \left\{\begin{array}{@{}cl} \displaystyle x^{\frac{5}{12}+\varepsilon}, & \ \mathrm{for} \ k=3, \\ \displaystyle x^{\frac12 -\frac{1}{2k}+\varepsilon}, & \ \mathrm{for} \ k=4,5,6,7,\ldots, \end{array} \right. \end{align} $$

where $\varepsilon $ denotes an arbitrary, small positive number that may be different at each occurrence.

For $k=3$ , the estimate (1.14) gives us an improvement on the estimate (1.9) under the RH.