On the distribution of \(p^\alpha\) modulo one
type: Item
instance of: scholarly article
title: On the distribution of \(p^\alpha\) modulo one
zbMATH Open document ID: 0988.11027
author: Xiaodong Cao
author: Wenguang Zhai
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 2000-04-03T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/93747
review text: The authors give a new upper bound for the discrepancy \[ D(N):= \sup_{0\leq\gamma\leq 1} \Bigl|\sum_{\substack{ p\leq N\\ \{p^\alpha\}\leq \gamma}} 1-\pi(N)\gamma \Bigr| \] for the sequence \((p^\alpha)\) when \(5/3\leq \alpha\leq 3\) and \(\alpha\neq 2\). As usual, \(p\) denotes a prime number, \(\pi(N)\) is the number of primes less than or equal to \(N\) and \(\{x\}\) is the fractional part of the real number \(x\). They prove that for the specified range of \(\alpha\) the estimate \(D(N)= O(N^{1-\delta+\varepsilon})\) holds, where \(\delta= \delta(\alpha)\) is explicitly given. They combine the double large sieve inequality due to Bombieri and Iwaniec and Heath-Brown's method to decompose sums over primes.
Mathematics Subject Classification ID: 11J71
Mathematics Subject Classification ID: 11L07
Mathematics Subject Classification ID: 11K06
Mathematics Subject Classification ID: 11N36
zbMATH DE Number: 1426379
zbMATH Keywords: upper bound
zbMATH Keywords: discrepancy
zbMATH Keywords: double large sieve inequality
zbMATH Keywords: sums over primes
reviewed by: Dieter Leitmann
MaRDI profile type: Publication
OpenAlex ID: W2056856135
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DOI: 10.5802/JTNB.258
description: scientific article; zbMATH DE number 1426379