On the mean values of the error terms in Mertens' theorems
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866911019937300480 |
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| author | Zhao, Tianyu |
| author_facet | Zhao, Tianyu |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_18903 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the mean values of the error terms in Mertens' theorems Zhao, Tianyu Number Theory 11M06, 11M26, 11N05, 11Y35 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. |
| title | On the mean values of the error terms in Mertens' theorems |
| topic | Number Theory 11M06, 11M26, 11N05, 11Y35 |
| url | https://arxiv.org/abs/2411.18903 |