Some estimates of trigonometric sums and their applications
type: Item
instance of: scholarly article
title: Some estimates of trigonometric sums and their applications
zbMATH Open document ID: 0906.11043
DOI: 10.1007/BF02907049
published in: Acta Mathematica Hungarica
publication date: 1999-01-24T00:00:00Z
review text: This paper investigates the general classical Kloosterman sum \[ R_{n,m}(u,v) = \sum_{\substack{ 1\leq a,b \leq n\\ ab\equiv 1\pmod n}} e\left({au+bv\over m}\right), \] where \(m,n,u,v\) are integers with \(m\geq 2\), \(n\geq 3\), and \(e(y) = e^{2\pi i y}\). When \(m=n\), this is the familiar Kloosterman sum which satisfies the Weil estimate \(\ll (u,v,n)^{1\over 2}n^{1\over 2}d(v)\), where \((u,v,n)\) denotes the greatest common divisor of \(u\), \(v\), and \(n\). The author shows that if \(m\neq n\), then when \(n\) is a prime one obtains an asymptotic formula for \(R\) with a main term (which vanishes if \(m=n\).) The result is that if \(p\) is an odd prime, and \(m,u,v\) are integers with \(m\nmid u\), \(n\nmid v\), then \[ R_{p,m}(u,v) = {1\over p}e\left({(u+v)(p+1)\over 2m}\right) {\sin{\pi p u\over m} \sin{\pi p v\over m}\over \sin{\pi u\over m}\sin{\pi v\over m}} +O(\sqrt{p}\log ^2p). \] The proof of this result is remarkably simple, with the main term arising naturally and the error terms estimated by the Weil estimate above. By the same method the author next examines the distribution of multiplicative inverses \(\overline{a}\) determined by \(a\overline{a} \equiv 1 \pmod n\) when \((a,n) = 1\). Define \[ S(n,k,\sigma) = \sideset{}{'}\sum_{\substack{ a=1\\ | a- \overline{a}| <\sigma n}}^n | a- \overline{a}| ^k, \] for \(k\geq 0\), \(0\leq \sigma \leq 1\), and \(\sum_a'\) indicates a sum over \(a\) with \((a,n)=1\). The author proves \[ S(n,k,\sigma)= 2\phi(n)n^k \left ({\sigma^{k+1}\over k+1}- {\sigma^{k+2}\over k+2}\right) + O\left(n^{k + {1\over 2}}d^2(n)\log ^2 n\right). \] He also obtains a similar result for \(-1<k<0\). Further, he proves that for \(p\) an odd prime, and \(m\) an integer with \(1<m<p\), then \[ \# \left\{a : 1\leq a\leq p-1, \;a\overline{a}\equiv 1 \pmod p, a\equiv \overline{a} \pmod m\right\} = \tfrac pm+ O(\sqrt{p}\log ^2 p). \] From these results a number of corollaries are obtained for sums involving \(a-\overline{a}\).
Mathematics Subject Classification ID: 11L05
Mathematics Subject Classification ID: 11N69
zbMATH DE Number: 1244248
zbMATH Keywords: Kloosterman sums
zbMATH Keywords: asymptotic formula
zbMATH Keywords: distribution of multiplicative inverses
author: Wenpeng Zhang
MaRDI profile type: Publication
cites work: On Kloosterman's sum
cites work: Kloosterman sums and Fourier coefficients of cusp forms
cites work: Q4101884
cites work: On the difference between an integer and its inverse modulo \(n\)
description: scientific article; zbMATH DE number 1244248