On the symmetry of primes

2010/09/29 by Giovanni Coppola, Coppola, Giovanni
Mathematics · #11N25 #11N37 #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11N25 #msc:11N37

paper · pdf · doi:10.48550/arxiv.1009.6121

The paper has been withdrawn by the Author see 1103.4451v2 comments

arxiv created 2011/05/30 · arxiv updated 2011/05/31

Abstract

We prove a kind of "almost all symmetry" result for the primes, i.e. we give non-trivial bounds for the "symmetry integral", say I_Λ(N,h), of the von Mangoldt function Λ(n) (:= log p for prime-powers n=p^r, 0 otherwise). We get I_Λ(N,h)≪ NhL⁵+Nh^21/20L², with L:=log N; as a Corollary, we bound non-trivially the Selberg integral of the primes, i.e. the mean-square of ∑_{x<n≤ x+h}Λ(n)-h, over x∈ [N,2N], to get the "Prime Number Theorem in almost all short intervals" of (log-powers!) length h≥ L^11/2+ε. We trust here in the improvement of the exponent, say c<11/2.

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