On a trigonometric sum and its application
type: Item
instance of: scholarly article
title: On a trigonometric sum and its application
zbMATH Open document ID: 0626.10039
DOI: 10.1007/BF01950995
author: N. A. Fugelo
author: P. D. Varbanets
published in: Acta Mathematica Hungarica
publication date: 1987-01-01T00:00:00Z
review text: The problem of distribution of square-free numbers in short intervals stands in a close connection with the problem of estimating the number of lattice points \(D(1,2;x)\) in the domain \(\xi \eta^2\leq x\), \(\xi >0\), \(\eta >0\). It is well-known that \[ D(1,2;x)=\zeta (2)x+\zeta (1/2)\sqrt{x}+\Delta (1,2;x), \] \[ \Delta (1,2;x)=\sum_{n^3\leq x}\{\psi (x/n^2)+\psi (\sqrt{x/n})\}+O(1) = O(x^{2/9} \log x), \] which was proved by \textit{H.-E. Richert} [Math. Z. 56, 21--32 (1952; Zbl 0046.25002)]. \textit{P. G. Schmidt} [Abschätzungen bei unsymmetrischen Gitterpunktproblemen (Diss. Göttingen, 1964)] found the improvement \(\Delta (1,2;x)\ll x^{0,2215\ldots}\). The most difficult problem is the estimation of the first sum of \(\Delta(1,2;x)\). The authors now apply the method of exponent pairs for double exponential sums, developed by \textit{B. R. Srinivasan} [Math. Ann. 160, 280--311 (1965; Zbl 0141.04603)], and prove the estimation \[ \sum_{n^3\leq x}\psi(x/n^2) \ll x^{0,21829}. \]
reviewed by: Ekkehard Krätzel
Mathematics Subject Classification ID: 11N37
Mathematics Subject Classification ID: 11L40
Mathematics Subject Classification ID: 11P21
zbMATH DE Number: 4017018
zbMATH Keywords: distribution of values of multiplicative functions
zbMATH Keywords: short intervals
zbMATH Keywords: distribution of square-free numbers
zbMATH Keywords: number of lattice points
zbMATH Keywords: exponent pairs
zbMATH Keywords: double exponential sums
MaRDI profile type: Publication
cites work: The lattice point problem of many dimensional hyperboloids. III
cites work: On the number of Abelian groups of a given order
description: scientific article; zbMATH DE number 4017018