The Cramér conjecture holds with a positive probability

Published 23 Feb 2018 in math.NT | (1802.08524v1)

Abstract: We prove that a positive proportion of the intervals of any fixed scalar multiple of $\log(X)$ in the dyadic interval $[X,2X]$ contain a prime number. We also show that a positive proportion of the congruence classes modulo $q$ contain a prime number smaller than any fixed scalar multiple of $\varphi(q)\log(q).$

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Naser T. Sardari

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