Exceptional characters and prime numbers in short intervals
type: Item
instance of: scholarly article
title: Exceptional characters and prime numbers in short intervals
zbMATH Open document ID: 1054.11048
published in: Selecta Mathematica. New Series
publication date: 2004-09-27T00:00:00Z
review text: In the second of their series of papers concerning the consequences for the distribution of primes which follow from the assumption of the existence of exceptional Dirichlet characters \(\chi\pmod D\), the authors deal with the problem of the distribution of primes in short intervals \(x-y<p\leq x\). The Riemann Hypothesis implies that \(\psi(x)-\psi(x-y)=y+O(x^{{1\over2}+\varepsilon})\), which has content for \(x^{{1\over2}+\varepsilon}<y<x\), and their goal is to reach a fixed power below \(\sqrt x\). The following is their main theorem: Let \(\chi\) be the real primitive character of conductor \(D\). Let \(x\geq D^r\), with \(r=18290\), and \(x^{39\over79}\leq y\leq x\). Then \[ \psi(x)-\psi(x-y)=y\{1+O(L(1,\chi)(\log x)^{r^r})\}, \] where \(L(s,\chi)\) is the Dirichlet \(L\)-function, and the implied constant is absolute and computable. Although the result here is unconditional, the asymptotic formula has meaning only when \(L(1,\chi)\) is exceptionally small, so that \(\chi\) is exceptional; thus, for an error term \(O(1/\log x)\) the requirement is \(L(1,\chi)\ll(\log x)^{-{r^r}-1}\). Moreover, because of the restriction on~\(D\) relative to~\(x\), the result does not hold for every interval, although there will be such intervals if \(L(1,\chi)\) is really that small. The proof follows the lines introduced in their first paper [Int. Math. Res. Not. 2003, No. 37, 2033--2050 (2003; Zbl 1038.11060)]. The method involves the arithmetic function \(\rho=\chi\star1\star1\); in particular the proof requires an asymptotic formula for the average value for \(\rho(n)\). The authors show that, in the presence of exceptional characters, it suffices to deal with some twist of the divisor function \(\tau_2(n)\) to obtain the main term, and a similar twist of \(\tau_3(n)\) to estimate the error term. Thus one may say that the possibility for an argument which leads to a result with a fixed power below \(\sqrt x\) follows from the fact that the error term associated with the a…
Mathematics Subject Classification ID: 11N05
Mathematics Subject Classification ID: 11M20
Mathematics Subject Classification ID: 11M26
zbMATH DE Number: 2103785
zbMATH Keywords: Primes
zbMATH Keywords: intervals
zbMATH Keywords: exceptional characters.
author: J. B. Friedlander
author: H. Iwaniec
reviewed by: Peter Shiu
MaRDI profile type: Publication
full work available at URL: s00029-004-0374-6
OpenAlex ID: W2018381070
DOI: 10.1007/S00029-004-0374-6
description: scientific article; zbMATH DE number 2103785