arXiv subjects
Publications and source records attributed to Zhai Wenguang.
Let $f$ be any arithmetic function and define $S_f(x):=\sum_{n\leq x}f([x/n])$. If the function $f$ is small, namely, $f(n)\ll n^\varepsilon,$ then the error term $E_f(x)$ in the asymptotic formula of $S_f(x)$ has the form $O(x^{1/2+\varepsilon}).$ In this paper, we shall study the mean square of $E_f(x)$ and establish some new results of $E_f(x)$ for some special functions.
Let ${\mathbb Z}_{n}$ be the additive group of residue classes modulo $n.$ For any positive integers $n_1$ and $n_2$, let $s(n_1,n_2)$ and $c(n_1,n_2)$ denote the number of subgroups and the number of cyclic subgroups of the group ${\mathbb Z}_{n_1}\times {\mathbb Z}_{n_2}$, respectively. The aim of this paper is to study the asymptotic behavior of the sums $\sum_{n_1,n_2\le x}s(n_1,n_2)$ and $\sum_{n_1,n_2\le x}c(n_1,n_2)$. Some sharper asymptotic results are given for the these sums. Mean values of the error terms are also studied.
This is a survey article on the moments of a class of error terms in analytic number theory. We begin by reviewing the moments of the Dirichlet divisor problem, and then review the hybrid moments of error terms in the divisor problems. We give two new results on the joint distribution of sign changes of $\Delta_2(x)$ and $\Delta_3(x). $ In the last section, we present some conjectures on the moment results in the well-known Selberg class.