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

Using a recent verification of the Riemann hypothesis up to height 3 · 10 12 , we provide strong estimates on π ( x ) and other prime counting functions for finite ranges of x . In particular, we get that | π ( x ) - l i ( x ) | < x log x / 8 π for 2657 ≤ x ≤ 1.101 · 10 26 . We also provide weake...

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Bibliographic Details
Published in:The Ramanujan journal 2022-12, Vol.59 (4), p.1307-1321
Main Author: Johnston, Daniel R.
Format: Article
Language:English
Subjects:
Combinatorics
Field Theory and Polynomials
Fourier Analysis
Functions of a Complex Variable
Hypotheses
Mathematics
Mathematics and Statistics
Number Theory
Verification
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Summary:Using a recent verification of the Riemann hypothesis up to height 3 · 10 12 , we provide strong estimates on π ( x ) and other prime counting functions for finite ranges of x . In particular, we get that | π ( x ) - l i ( x ) | < x log x / 8 π for 2657 ≤ x ≤ 1.101 · 10 26 . We also provide weaker bounds that hold for a wider range of x , and an application to an inequality of Ramanujan.
ISSN:1382-4090
1572-9303
DOI:10.1007/s11139-022-00616-x