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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| Vydáno v: | Research in number theory Ročník 8; číslo 4 |
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| Médium: | Journal Article |
| Jazyk: | angličtina |
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Springer International Publishing
01.12.2022
Springer Nature B.V |
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| ISSN: | 2522-0160, 2363-9555 |
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| Abstract | 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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| AbstractList | 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. |
| ArticleNumber | 71 |
| Author | Goldston, D. A. Suriajaya, Ade Irma |
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| Cites_doi | 10.4064/aa-37-1-339-343 10.1007/BF01444344 10.4064/aa-41-1-85-99 10.1007/BF02403071 10.1007/s00208-008-0280-x 10.1007/BF01475582 10.1007/978-1-4612-4816-3_10 10.1090/pspum/024/9944 |
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| Copyright | The Author(s), under exclusive licence to Springer Nature Switzerland AG 2022. Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. |
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| Keywords | 11M26 Prime numbers 11M06 11N05 Riemann zeta-function Prime number theorem |
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| References | Schmidt (CR13) 1903; 57 CR4 CR5 Goldston, Heath-Brown (CR6) 1984; 266 CR9 Ford, Soundararajan, Zaharescu (CR1) 2009; 343 Montgomery, Vaughan (CR11) 2007 CR14 Heath-Brown (CR7) 1982; 41 CR12 Gallagher (CR2) 1980; 37 Hugh (CR8) 1971 CR10 von Koch (CR15) 1901; 24 Gallagher, Mueller (CR3) 1978; 303–304 PX Gallagher (371_CR2) 1980; 37 PX Gallagher (371_CR3) 1978; 303–304 E Schmidt (371_CR13) 1903; 57 371_CR5 371_CR4 HL Montgomery (371_CR11) 2007 371_CR12 371_CR10 L Hugh (371_CR8) 1971 H von Koch (371_CR15) 1901; 24 371_CR14 K Ford (371_CR1) 2009; 343 DA Goldston (371_CR6) 1984; 266 371_CR9 DR Heath-Brown (371_CR7) 1982; 41 |
| References_xml | – year: 1971 ident: CR8 publication-title: Montgomery, Topics in Multiplicative Number. Theory Lecture Notes in Mathematics – volume: 37 start-page: 339 year: 1980 end-page: 343 ident: CR2 article-title: Some consequences of the Riemann hypothesis publication-title: Acta Arith. doi: 10.4064/aa-37-1-339-343 – volume: 57 start-page: 195 issue: 2 year: 1903 end-page: 204 ident: CR13 article-title: Über die Anzahl der Primzahlen unter gegebener Grenze publication-title: Math. Ann. doi: 10.1007/BF01444344 – ident: CR4 – ident: CR14 – ident: CR12 – volume: 41 start-page: 85 year: 1982 end-page: 99 ident: CR7 article-title: Gaps between primes, and the pair correlation of zeros of the zeta-function publication-title: Acta Arith. doi: 10.4064/aa-41-1-85-99 – volume: 24 start-page: 159 year: 1901 end-page: 182 ident: CR15 article-title: Sur la distribution des nombres premiers publication-title: Acta Math. doi: 10.1007/BF02403071 – ident: CR10 – year: 2007 ident: CR11 publication-title: Multiplicative Number Theory, Cambridge Studies in Advanced Mathematics – ident: CR9 – volume: 303–304 start-page: 205 year: 1978 end-page: 220 ident: CR3 article-title: Primes and zeros in short intervals publication-title: J. Reine Angew. Math. – ident: CR5 – volume: 343 start-page: 487 year: 2009 end-page: 505 ident: CR1 article-title: On the distribution of imaginary parts of zeros of the Riemann zeta function, II publication-title: Math. Ann. doi: 10.1007/s00208-008-0280-x – volume: 266 start-page: 317 year: 1984 end-page: 320 ident: CR6 article-title: A note on the differences between consecutive primes publication-title: Math. Ann. doi: 10.1007/BF01475582 – volume-title: Montgomery, Topics in Multiplicative Number. Theory Lecture Notes in Mathematics year: 1971 ident: 371_CR8 – ident: 371_CR12 – ident: 371_CR9 – ident: 371_CR14 – volume: 24 start-page: 159 year: 1901 ident: 371_CR15 publication-title: Acta Math. doi: 10.1007/BF02403071 – ident: 371_CR4 doi: 10.1007/978-1-4612-4816-3_10 – volume: 37 start-page: 339 year: 1980 ident: 371_CR2 publication-title: Acta Arith. doi: 10.4064/aa-37-1-339-343 – ident: 371_CR5 – volume: 343 start-page: 487 year: 2009 ident: 371_CR1 publication-title: Math. Ann. doi: 10.1007/s00208-008-0280-x – volume: 57 start-page: 195 issue: 2 year: 1903 ident: 371_CR13 publication-title: Math. Ann. doi: 10.1007/BF01444344 – volume: 41 start-page: 85 year: 1982 ident: 371_CR7 publication-title: Acta Arith. doi: 10.4064/aa-41-1-85-99 – volume: 266 start-page: 317 year: 1984 ident: 371_CR6 publication-title: Math. Ann. doi: 10.1007/BF01475582 – ident: 371_CR10 doi: 10.1090/pspum/024/9944 – volume: 303–304 start-page: 205 year: 1978 ident: 371_CR3 publication-title: J. Reine Angew. Math. – volume-title: Multiplicative Number Theory, Cambridge Studies in Advanced Mathematics year: 2007 ident: 371_CR11 |
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| Title | The prime number theorem and pair correlation of zeros of the Riemann zeta-function |
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