View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by PORTO Publications Open Repository TOrino Prime numbers in logarithmic intervals Danilo Bazzanella Dipartimento di Matematica, Politecnico di Torino, Italy
[email protected] Alessandro Languasco Dipartimento di Matematica Pura e Applicata Universit`adi Padova , Italy
[email protected] Alessandro Zaccagnini Dipartimento di Matematica Universit`adi Parma, Italy
[email protected] First published in Transactions of the American Mathematical Society 362 (2010), no. 5, 2667-2684 published by the American Mathematical Society DOI: 10.1090/S0002-9947-09-05009-0 The original publication is available at http://www.ams.org/ 1 TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 362, Number 5, May 2010, Pages 2667–2684 S0002-9947(09)05009-0 Article electronically published on November 17, 2009 PRIME NUMBERS IN LOGARITHMIC INTERVALS DANILO BAZZANELLA, ALESSANDRO LANGUASCO, AND ALESSANDRO ZACCAGNINI 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 X is a prime number and h = o(X). Then we will apply this to prove that ≤ for every λ>1/2thereexistsapositiveproportionofprimesp X such that ≤ the interval (p, p+λ log X]containsatleastaprimenumber.Asaconsequence we improve Cheer and Goldston’s result on the size of real numbers λ> 1 with the property that there is a positive proportion of integers m X ≤ such that the interval