On the nth record gap between primes in an arithmetic progression

Autor:	Alexei Kourbatov
Rok vydání:	2018
Předmět:	Conjecture Mathematics - Number Theory Coprime integers Distribution (number theory) Quadratic function 11N05 11N13 11N56 Upper and lower bounds Combinatorics Gumbel distribution Arithmetic progression Log-normal distribution FOS: Mathematics Number Theory (math.NT) Mathematics
Zdroj:	International Mathematical Forum. 13:65-78
ISSN:	1314-7536
DOI:	10.12988/imf.2018.712103
Popis:	Let $q>r\ge1$ be coprime integers. Let $R(n,q,r)$ be the $n$th record gap between primes in the arithmetic progression $r$, $r+q$, $r+2q,\ldots,$ and denote by $N_{q,r}(x)$ the number of such records observed below $x$. For $x\to\infty$, we heuristically argue that if the limit of $N_{q,r}(x)/\log x$ exists, then the limit is 2. We also conjecture that $R(n,q,r)=O_q(n^2)$. Numerical evidence supports the conjectural (a.s.) upper bound $$R(n,q,r) Comment: 14 pages, 4 figures, 2 tables. Sequel to arXiv:1610.03340
Databáze:	OpenAIRE
Externí odkaz:	https://explore.openaire.eu/search/publication?articleId=doi_dedup___::6739481a406328fa14f401b006f814d7 https://doi.org/10.12988/imf.2018.712103 Zobrazit plný text záznamu