Estimate for exponential sums and its applications
type: Item
instance of: scholarly article
title: Estimate for exponential sums and its applications
zbMATH Open document ID: 1314.11052
author: Weili Yao
published in: Frontiers of Mathematics in China
publication date: 2012-11-07T00:00:00Z
review text: Let \(f_{k}(n)\) be the characteristic function of \(n\) with \(\Omega(n)=k\), and \[ T_{k}(x,\alpha)=\sum_{n\leq x}f_{k}(n)e(n\alpha). \] Write \[ r(N)=\#\{N=f_{k}(n_{1})n_{1}+f_{k}(n_{2})n_{2}+f_{k}(n_{3})n_{3}\}. \] The author obtains the following result: Let \(k\leq (\log\log N)^{1-\Delta}\), then \[ r(N)=\frac{1}{2}N^{2}\mathfrak{S}(N)(\frac{(\log\log N)^{k-1}}{(k-1)!\log N})^{3}(1+O(\frac{1}{(\log\log N)^{\Delta}})), \] where \(\mathfrak{S}(N)\) is the singular series in the ternary Goldbach problem. The proof uses the circle method and a Bombieri-type estimate for the exponential sum \(T_{k}(x,\alpha)\).
reviewed by: Zaizhao Meng
Mathematics Subject Classification ID: 11L07
Mathematics Subject Classification ID: 11M20
Mathematics Subject Classification ID: 11P32
zbMATH DE Number: 6103008
zbMATH Keywords: exponential sum
zbMATH Keywords: zero-density estimate
zbMATH Keywords: circle method
MaRDI profile type: Publication
full work available at URL: s11464-012-0222-5
OpenAlex ID: W2163188967
cites work: Q4652230
cites work: Q4023619
cites work: Q5301977
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cites work: On the distribution of integers with a fixed number of prime factors
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DOI: 10.1007/S11464-012-0222-5
description: scientific article; zbMATH DE number 6103008