Omega theorems for divisor functions
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, 699-721 (1961; Zbl 0100.039)], \textit{K. Corradi} and \textit{I. Kátai} [Magyar Tud. Akad. Mat. Fiz. Tud. Oszt. Közl. 17, 89-97 (1967; Zbl 0163.041)], \textit{H. Joris} [Comment. Math. Helv. 47, 220-248 (1972; Zbl 0248.12009) and ibid. 48, 409-435 (1973; Zbl 0282.12009)] and \textit{D. Redmond} [Rocky Mt. J. Math. 9, 733-748 (1979; Zbl 0388.10027)]. However, there are substantially new technical difficulties to be overcome, which is done by the author in an ingenious way. As corollaries, \(\Omega\)-results for the above-defined function R(x,r) are obtained. The reviewer feels that this paper is an outstanding contribution to our present knowledge about the theory of divisor functions (and related types of arithmetical functions) both for the wealth of special results and for the above-mentioned general \(\Omega\)-theorem which promisses to be applicable to a great number of related number-theoretic problems.
scientific article; zbMATH DE number 3886006