On the divisor and circle problems
type: Item
instance of: scholarly article
title: On the divisor and circle problems
zbMATH Open document ID: 0644.10031
DOI: 10.1016/0022-314X(88)90093-5
author: Charles J. Mozzochi
author: H. Iwaniec
published in: Journal of Number Theory
publication date: 1988-01-01T00:00:00Z
review text: This remarkable paper contains the estimate \(\Delta (x)\ll x^{7/22+\epsilon}\), and \(P(x)\ll x^{7/22+\epsilon}\), where as usual \[ \Delta (x)=\sum_{n\leq x}d(n)-x(\log x+2\gamma -1),\quad P(x)=\sum_{n\leq x}r(n)-\pi x \] denote the error terms in the divisor and circle problem, respectively. The exponent \(7/22=0.3181818...\) improves the previously best known exponent \(139/429=0.324009324...,\) which is due to \textit{G. Kolesnik} [Acta Arith. 45, 115-143 (1985; Zbl 0571.10036)]. Although the improvement on the value of the exponent is substantial, even more important is the wealth of ideas that are used in the paper. The details of the proof are given only for \(\Delta\) (x), since some slight modifications of the authors' arguments also work for the closely related circle problem (for this and a discussion of the history of estimates for \(\Delta(x)\) and P(x) see Ch. 13 of the reviewer's book [``Riemann zeta-function'' (Wiley 1985; Zbl 0556.10026)]). Instead of treating \(\Delta(x)\) by (double) exponential sums via Voronoi's formula, as was done by several generations of number theorists from E. C. Titchmarsh to G. Kolesnik, the authors adopt a new approach. They start from the elementary formula \[ (1)\quad \Delta (x)=- 2\sum_{n\leq \sqrt{x}}\psi (x/n)+0(1),\quad \psi (t)=t-[t]-1/2 \] and use a variety of techniques. These include approximation by Farey points and ideas from the Weyl and van der Corput treatment of exponential sums, such as Poisson summation. The (technically quite complicated) paper draws on ideas from the pioneering work of \textit{E. Bombieri} and the first author [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 13, 473-486 (1986; Zbl 0615.10046) and ibid. Ser. 13, 449-472 (1986; Zbl 0615.10047)] in which they proved \(\zeta (1/2+it)=O(t^{9/56+\epsilon}).\) Note that \(7/22<2\cdot (9/56)=0.3214285...,\) so that in a certain sense the methods of the present paper are even more successful, because one generally expects the exponent that on…
Mathematics Subject Classification ID: 11N37
Mathematics Subject Classification ID: 11P21
Mathematics Subject Classification ID: 11L40
zbMATH DE Number: 4049730
zbMATH Keywords: error terms
zbMATH Keywords: circle problem
zbMATH Keywords: approximation by Farey points
zbMATH Keywords: exponential sums
zbMATH Keywords: Poisson summation
zbMATH Keywords: divisor problem
zbMATH Keywords: incomplete theta-series
zbMATH Keywords: estimation of exponential sums
zbMATH Keywords: exponent pairs
zbMATH Keywords: estimation of lattice points
zbMATH Keywords: planar regions
Wikidata QID: Q30053046
reviewed by: Aleksandar Ivić
MaRDI profile type: Publication
cites work: Q4723831
cites work: Q5621389
cites work: Q4275389
cites work: Q4692762
cites work: A zero-density theorem for the Riemann zeta-function
cites work: On the method of exponent pairs
cites work: Q5803461
cites work: Q5846799
description: scientific article; zbMATH DE number 4049730