A note on large gaps between consecutive primes in arithmetic progressions
type: Item
instance of: scholarly article
title: A note on large gaps between consecutive primes in arithmetic progressions
zbMATH Open document ID: 0770.11042
DOI: 10.1016/0022-314X(92)90111-2
author: Alessandro Zaccagnini
published in: Journal of Number Theory
publication date: 1993-05-16T00:00:00Z
review text: Let \((a,q)=1\) and let \(G(x;a,q)\) denote the maximum of \(p_{n+1}-p_ n\), taken over consecutive primes \(p_ n\), \(p_{n+1}\equiv a\pmod q\), with \(p_ n\leq x\). Then for any constant \(C>0\) it is shown that \[ G(qx;q,a)\geq(e^ \gamma+o_ C(1))\varphi(q)\log x\log_ 2{\log_ 4x\over(\log_ 3x)^ 2}, \] uniformly for \(\omega(q)\leq\{C\log_ 2x(\log_ 4x)/(\log_ 3x)\}\). Here \(\log_ n\) denotes the iterated logarithm as usual. The case \(q=1\) is the well-known result of \textit{R. A. Rankin} [Proc. Edinb. Math. Soc., II. Ser. 13, 331-332 (1963; Zbl 0121.047)]. The proof depends on a variant of a result of \textit{C. Pomerance} [J. Number Theory 12, 218-223 (1980; Zbl 0436.10020)].
Mathematics Subject Classification ID: 11N13
zbMATH DE Number: 166050
zbMATH Keywords: large gaps between consecutive primes
zbMATH Keywords: arithmetic progressions
zbMATH Keywords: uniform estimate
reviewed by: D. R. Heath-Brown
MaRDI profile type: Publication
cites work: Q3275022
cites work: Unusually Large Gaps Between Consecutive Primes
cites work: A note on the least prime in an arithmetic progression
cites work: The Difference between Consecutive Prime Numbers V
Recommended article: Q3754682
Recommended article: A remark on large gaps between primes in arithmetic progressions
Recommended article: A note on small gaps between primes in arithmetic progressions
Recommended article: Q5425794
Recommended article: On the gaps between consecutive primes
Recommended article: Some new results on gaps between consecutive primes
Recommended article: Large prime gaps and progressions with few primes
Recommended article: Q5371248
Recommended article: Q2967789
Recommended article: On the Ratio of Consecutive Gaps Between Primes
description: scientific article; zbMATH DE number 166050