Improving bounds on prime counting functions by partial verification of the Riemann hypothesis

DOI10.1007/S11139-022-00616-XarXiv2109.02249WikidataQ114223553 ScholiaQ114223553MaRDI QIDQ6376926

D. Johnston

Publication date: 6 September 2021

Abstract: Using a recent verification of the Riemann hypothesis up to height $3cdot10^{12}$, we provide strong estimates on $pi(x)$ and other prime counting functions for finite ranges of $x$. In particular, we get that $|pi(x)-extli(x)|<sqrtxlogx/8pi$ for $2657leqxleq1.101cdot10^{26}$. We also provide weaker bounds that hold for a wider range of $x$, and an application to an inequality of Ramanujan.

Mathematics Subject Classification ID

Nonreal zeros of (zeta (s)) and (L(s, chi)); Riemann and other hypotheses (11M26) Distribution of primes (11N05) Values of arithmetic functions; tables (11Y70)

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