The prime number theorem and pair correlation of zeros of the Riemann zeta-function

We prove that the error in the prime number theorem can be quantitatively improved beyond the Riemann Hypothesis bound by using versions of Montgomery’s conjecture for the pair correlation of zeros of the Riemann zeta-function which are uniform in long ranges and with suitable error terms.

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Bibliographic Details
Published in:Research in number theory Vol. 8; no. 4
Main Authors: Goldston, D. A., Suriajaya, Ade Irma
Format: Journal Article
Language:English
Published: Cham Springer International Publishing 01.12.2022
Springer Nature B.V
Subjects:
Mathematics
Mathematics and Statistics
Number Theory
Prime numbers
Theorems
11M26
Prime numbers
11M06
11N05
Riemann zeta-function
Prime number theorem
ISSN:2522-0160, 2363-9555
Online Access:Get full text
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Summary:We prove that the error in the prime number theorem can be quantitatively improved beyond the Riemann Hypothesis bound by using versions of Montgomery’s conjecture for the pair correlation of zeros of the Riemann zeta-function which are uniform in long ranges and with suitable error terms.
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ISSN:2522-0160
2363-9555
DOI:10.1007/s40993-022-00371-4