On the mean values of the error terms in Mertens' theorems

2024/11/28 by Zhao, Tianyu
#11M06 #11M26 #11N05 #11Y35 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2411.18903

Abstract

For i∈ \1,2,3\, let E_i(x) denote the error term in each of the three theorems of Mertens on the asymptotic distribution of prime numbers. We show that for i∈ \1,2\ the Riemann hypothesis is equivalent to the condition ∫₂^X E_i(x) dx>0 for all X>2, and we examine assumptions under which the equivalence also holds for i=3. In addition, we extend our results to analogues of Mertens' theorems concerning prime sums twisted by quadratic Dirichlet characters or restricted to arithmetic progressions.

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