Prime numbers in logarithmic intervals
Authors:
Danilo Bazzanella, Alessandro Languasco and Alessandro Zaccagnini
Journal:
Trans. Amer. Math. Soc. 362 (2010), 26672684
MSC (2010):
Primary 11N05; Secondary 11A41
Published electronically:
November 17, 2009
MathSciNet review:
2584615
Fulltext PDF Free Access
Abstract 
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Abstract: Let be a large parameter. We will first give a new estimate for the integral moments of primes in short intervals of the type , where is a prime number and . Then we will apply this to prove that for every there exists a positive proportion of primes such that the interval contains at least a prime number. As a consequence we improve Cheer and Goldston's result on the size of real numbers with the property that there is a positive proportion of integers such that the interval contains no primes. We also prove other results concerning the moments of the gaps between consecutive primes and about the positive proportion of integers such that the interval 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 such that the interval contains no primes.
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Gaps between primes, and the pair correlation of zeros of the zetafunction. Acta Arith., 41:8599, 1982. MR 667711 (83m:10078)
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On the difference between consecutive primes. Invent. Math., 15:155164, 1972. MR 0292774 (45:1856)
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Additional Information
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
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
DOI:
http://dx.doi.org/10.1090/S0002994709050090
Received by editor(s):
September 17, 2008
Published electronically:
November 17, 2009
Article copyright:
© Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
