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arXiv · 2411.18903

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

Tianyu Zhao
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Abstract

For $i\in \{1,2,3\}$, 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 $i\in \{1,2\}$ the Riemann hypothesis is equivalent to the condition $\int_2^X E_i(x) \:\mathrm{d}x>0$ 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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Tianyu Zhao. 2025-06-24. On the mean values of the error terms in Mertens' theorems. https://doi.org/10.1007/s40993-025-00640-y

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