On the prime ideal theorem and irregularities in the distribution of primes
type: Item
instance of: scholarly article
title: On the prime ideal theorem and irregularities in the distribution of primes
zbMATH Open document ID: 0818.11035
DOI: 10.1215/S0012-7094-95-07701-1
published in: Duke Mathematical Journal
publication date: 1995-08-24T00:00:00Z
review text: For any fixed \(k\in \mathbb{N}\), let \(f(x)\) denote the polynomial \(x^ k+ Qd\) with \(Q,d\in \mathbb{N}\). Put, for fixed \(Q\) and prime \(p\), \[ \rho(d,p)= |\{m \bmod p;\;m^ k+ Qd\equiv 0\bmod p\}|. \] We factorize \(f\) over \(\mathbb{Q}\) as \(f= \prod_{i=1}^{r(d)} f_ i\), where each \(f_ i\) is irreducible and has integral coefficients, then the prime ideal theorem implies that \(\sum_{p\leq x} \rho(d, p)\sim r(d)\cdot \text{li } x\), \(x\to \infty\). In the present paper the authors derive the upper bound \[ \sum_{Y\leq d\leq Y+H} \;\biggl| \sum_{X\leq p\leq 2X} (\rho(d, p)- r(d)) \biggr| \ll_{A,k, \varepsilon} {{X\cdot H} \over {\log^ A X}} \] uniformly for \(Q\leq Y^ A\), \(A>0\), provided that \(Y^{(1/2)+ \varepsilon} \leq H\leq Y\), \(\varepsilon>0\). This result has an interesting application concerning a theorem of Friedlander-Granville [\textit{J. Friedlander} and \textit{A. Granville}, Proc. R. Soc. Lond., Ser. A 435, 197-204 (1991; Zbl 0736.11049)] on irregularities in the distribution of primes represented by polynomials.
Mathematics Subject Classification ID: 11N32
Mathematics Subject Classification ID: 11R42
zbMATH DE Number: 755402
zbMATH Keywords: Friedlander-Granville theorem
zbMATH Keywords: prime ideal theorem
zbMATH Keywords: upper bound
zbMATH Keywords: irregularities in the distribution of primes represented by polynomials
author: M. Nair
author: A. Perelli
reviewed by: Jürgen Hinz
MaRDI profile type: Publication
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full work available at URL: s0012-7094-95-07701-1
OpenAlex ID: W1479851868
description: scientific article; zbMATH DE number 755402