   ISSN 1088-6850(online) ISSN 0002-9947(print)

Prime numbers in logarithmic intervals

Authors: Danilo Bazzanella, Alessandro Languasco and Alessandro Zaccagnini
Journal: Trans. Amer. Math. Soc. 362 (2010), 2667-2684
MSC (2010): Primary 11N05; Secondary 11A41
DOI: https://doi.org/10.1090/S0002-9947-09-05009-0
Published electronically: November 17, 2009
MathSciNet review: 2584615
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Abstract: Let $X$ be a large parameter. We will first give a new estimate for the integral moments of primes in short intervals of the type $(p,p+h]$, where $p\leq X$ is a prime number and $h=o(X)$. Then we will apply this to prove that for every $\lambda >1/2$ there exists a positive proportion of primes $p\leq X$ such that the interval $(p,p+ \lambda \log X]$ contains at least a prime number. As a consequence we improve Cheer and Goldston’s result on the size of real numbers $\lambda >1$ with the property that there is a positive proportion of integers $m\leq X$ such that the interval $(m,m+ \lambda \log X]$ contains no primes. We also prove other results concerning the moments of the gaps between consecutive primes and about the positive proportion of integers $m\leq X$ such that the interval $(m,m+ \lambda \log X]$ contains at least a prime number. The last applications of these techniques are two theorems (the first one unconditional and the second one in which we assume the validity of the Riemann Hypothesis and of a form of the Montgomery pair correlation conjecture) on the positive proportion of primes $p\leq X$ such that the interval $(p,p+ \lambda \log X]$ contains no primes.

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Danilo Bazzanella
Affiliation: Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy
Email: danilo.bazzanella@polito.it

Alessandro Languasco
Affiliation: Dipartimento di Matematica Pura e Applicata, Università di Padova, Via Trieste 63, 35121 Padova, Italy
MR Author ID: 354780
ORCID: 0000-0003-2723-554X
Email: languasco@math.unipd.it

Alessandro Zaccagnini
Affiliation: Dipartimento di Matematica, Università di Parma, Parco Area delle Scienze, 53/a, Campus Universitario, 43100 Parma, Italy
Email: alessandro.zaccagnini@unipr.it

Received by editor(s): September 17, 2008
Published electronically: November 17, 2009