The mean square of the error term in the prime number theorem
| Autor: | Richard P. Brent, David J. Platt, Tim Trudgian |
|---|---|
| Rok vydání: | 2022 |
| Předmět: |
Mean square
Algebra and Number Theory Mathematics - Number Theory 010102 general mathematics 01 natural sciences Term (time) Combinatorics Riemann hypothesis symbols.namesake 0103 physical sciences FOS: Mathematics symbols 11M06 11M26 11N05 Number Theory (math.NT) 010307 mathematical physics Limit (mathematics) 0101 mathematics Prime number theorem Mathematics |
| Zdroj: | Journal of Number Theory. 238:740-762 |
| ISSN: | 0022-314X |
| DOI: | 10.1016/j.jnt.2021.09.016 |
| Popis: | We show that, on the Riemann hypothesis, $\limsup_{X\to\infty}I(X)/X^{2} \leq 0.8603$, where $I(X) = \int_X^{2X} (\psi(x)-x)^2\,dx.$ This proves (and improves on) a claim by Pintz from 1982. We also show unconditionally that $\frac{1}{5\,374}\leq I(X)/X^2 $ for sufficiently large $X$, and that the $I(X)/X^{2}$ has no limit as $X\rightarrow\infty$. Comment: 23 pages |
| Databáze: | OpenAIRE |
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