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AN EXPLICIT MEAN-VALUE ESTIMATE FOR THE PRIME NUMBER THEOREM IN INTERVALS

Published online by Cambridge University Press:  19 September 2023

MICHAELA CULLY-HUGILL  and
ADRIAN W. DUDEK
MICHAELA CULLY-HUGILL*
Affiliation:
School of Science, UNSW Canberra, Canberra ACT 2612, Australia
ADRIAN W. DUDEK
Affiliation:
School of Mathematics and Physics, University of Queensland, St Lucia, QLD 4072, Australia e-mail: a.dudek@uq.edu.au
*
e-mail: m.cully-hugill@adfa.edu.au
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Abstract

This paper gives an explicit version of Selberg’s mean-value estimate for the prime number theorem in intervals, assuming the Riemann hypothesis [25]. Two applications are given to short-interval results for primes and for Goldbach numbers. Under the Riemann hypothesis, we show there exists a prime in $(y,y+32\,277\log ^2 y]$ for at least half the $y\in [x,2x]$ for all $x\geq 2$, and at least one Goldbach number in $(x,x+9696 \log ^2 x]$ for all $x\geq 2$.

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Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 117 , Issue 1 , August 2024 , pp. 1 - 15
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

Communicated by Michael Coons

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