Omega theorems for divisor functions
type: Item
instance of: scholarly article
title: Omega theorems for divisor functions
zbMATH Open document ID: 0556.10031
author: Shigeru Kanemitsu
published in: Tokyo Journal of Mathematics
publication date: 1984-01-01T00:00:00Z
review text: Define for \(a\in {\mathbb{R}}\), \(k\in {\mathbb{N}}\), \(x\in {\mathbb{R}}^+\) \(G_{a,k}(x):=\sum_{n\leq \sqrt{x}}n^ a P_ k(x/n)\) where \(P_ k\) is the k-th periodic Bernoulli function. These functions arise as generalizations of \(G_{0,1}(x)\) which is connected with the error term \(\Delta\) (x) in the classical Dirichlet's divisor problem by the well- known relation (1) \(\Delta (x)=-2 G_{0,1}(x)+O(1)\) and have attracted a lot of attention in recent times: See \textit{S. Chowla} and \textit{H. Walum} [Proc. Symp. Pure Math. 8, 138-143 (1965; Zbl 0136.337)], the author and \textit{R. Sitaramachandrarao} [J. Number Theory (to appear) and ibid. 20, 103-120 (1985; Zbl 0467.10031 and Zbl 0467.10032)], the reviewer [Bull. Number Theory Rel. Top. 7, No.1, 1-10 (1982; Zbl 0491.10013)] and \textit{W. Recknagel} [Arch. Math. 44, 348-354 (1985; see the following review)]. As a first main result of the paper under review, a relation between \(G_{a,k}(x)\) and the remainder term R(x,r) in the asymptotic formula \[ \sum_{n\leq x}(x^ r-n^ r)\sigma_{-r}(n)=\frac{r}{1-r} \zeta (1- r)x+\frac{r}{1+r} \zeta (1+r)x^{r+1}-\frac{1}{2}\zeta (r)x^ r+\frac{1}{2}\zeta (-r)+R(x,r) \] (where \(\sigma_{-r}(n)=\sum_{d| n}d^{-r})\) is established: Theorem 1. For \(-1/2\leq r<3\), \(r\neq 0\), \(r\neq 1\), \[ R(x,r)=-(r/2)x^{r-1}G_{1-r,2}(x)+O(x^{(2r-1)/4}(\log x)^{\delta_{r,-1/2}}). \] (\(\delta\) is Kronecker's symbol.) As a consequence, upper bounds for R(x,r) are obtained on the basis of results from the above-cited papers. The second part of this work gives a general \(\Omega\)-estimate for the error term in the asymptotic formula for the summatory function of the coefficients of certain Dirichlet series which satisfy a Chandrasekharan- Narasimhan type functional equation [cf., (*) \textit{K. Chandrasekharan} and \textit{R. Narasimhan}, Ann. Math., II. Ser. 76, 93-136 (1962; Zbl 0211.379)]. The argument is in the spirit of (*), \textit{K. S. Gangadharan} [Proc. Camb. Philos. Soc. 57, 69…
Mathematics Subject Classification ID: 11N37
Mathematics Subject Classification ID: 11B39
zbMATH DE Number: 3886006
zbMATH Keywords: Bernoulli polynomial
zbMATH Keywords: Dirichlet's divisor problem
zbMATH Keywords: remainder term
zbMATH Keywords: asymptotic formula
zbMATH Keywords: upper bounds
zbMATH Keywords: \(\Omega \) -estimate
zbMATH Keywords: divisor functions
zbMATH Keywords: arithmetical functions
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.3836/tjm/1270151735
OpenAlex ID: W2083152074
DOI: 10.3836/TJM/1270151735
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description: scientific article; zbMATH DE number 3886006