Universality in Short Intervals of the Riemann Zeta-Function Twisted by Non-Trivial Zeros
Let 0<γ1<γ2<⋯⩽γk⩽⋯ be the sequence of imaginary parts of non-trivial zeros of the Riemann zeta-function ζ(s). Using a certain estimate on the pair correlation of the sequence {γk} in the intervals [N,N+M] with N1/2+ε⩽M⩽N, we prove that the set of shifts ζ(s+ihγk), h>0, approximating any...
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| Published in: | Mathematics (Basel) Vol. 8; no. 11; p. 1936 |
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| Language: | English |
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01.11.2020
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| Abstract | Let 0<γ1<γ2<⋯⩽γk⩽⋯ be the sequence of imaginary parts of non-trivial zeros of the Riemann zeta-function ζ(s). Using a certain estimate on the pair correlation of the sequence {γk} in the intervals [N,N+M] with N1/2+ε⩽M⩽N, we prove that the set of shifts ζ(s+ihγk), h>0, approximating any non-vanishing analytic function defined in the strip {s∈C:1/2<Res<1} with accuracy ε>0 has a positive lower density in [N,N+M] as N→∞. Moreover, this set has a positive density for all but at most countably ε>0. The above approximation property remains valid for certain compositions F(ζ(s)). |
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| AbstractList | Let 0<γ1<γ2<⋯⩽γk⩽⋯ be the sequence of imaginary parts of non-trivial zeros of the Riemann zeta-function ζ ( s ) . Using a certain estimate on the pair correlation of the sequence { γk } in the intervals [N,N+M] with N1/2+ε⩽M⩽N , we prove that the set of shifts ζ ( s+ihγk ) , h>0 , approximating any non-vanishing analytic function defined in the strip { s∈ C :1/2<Res<1 } with accuracy ε>0 has a positive lower density in [N,N+M] as N→∞ . Moreover, this set has a positive density for all but at most countably ε>0 . The above approximation property remains valid for certain compositions F ( ζ ( s ) ) . Let 0<γ1<γ2<⋯⩽γk⩽⋯ be the sequence of imaginary parts of non-trivial zeros of the Riemann zeta-function ζ(s). Using a certain estimate on the pair correlation of the sequence {γk} in the intervals [N,N+M] with N1/2+ε⩽M⩽N, we prove that the set of shifts ζ(s+ihγk), h>0, approximating any non-vanishing analytic function defined in the strip {s∈C:1/2<Res<1} with accuracy ε>0 has a positive lower density in [N,N+M] as N→∞. Moreover, this set has a positive density for all but at most countably ε>0. The above approximation property remains valid for certain compositions F(ζ(s)). |
| Author | Laurinčikas, Antanas Šiaučiūnas, Darius |
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| Cites_doi | 10.1007/s11139-017-9886-5 10.1515/crll.1914.144.249 10.3846/mma.2020.10450 10.1007/978-94-017-2091-5 10.1016/j.jnt.2019.04.006 10.1515/ms-2017-0141 10.1007/s00013-016-0998-8 10.4064/aa8583-5-2017 10.1007/BFb0060851 10.1007/BF01224983 10.1515/9783110886146 10.1090/pspum/024 10.1070/SM9194 |
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| References | (ref_10) 2018; 45 (ref_12) 2017; 181 ref_13 ref_23 ref_22 (ref_15) 2019; 210 ref_21 ref_20 (ref_11) 2018; 68 (ref_9) 2017; 108 ref_3 ref_19 ref_18 ref_17 Voronin (ref_2) 1975; 9 Reich (ref_7) 1980; 34 (ref_16) 2019; 204 (ref_8) 2019; 30 Bohr (ref_1) 1914; 144 ref_5 ref_4 (ref_14) 2020; 25 ref_6 |
| References_xml | – ident: ref_6 – volume: 45 start-page: 181 year: 2018 ident: ref_10 article-title: Joint universality for dependent L-functions publication-title: Ramanujan J. doi: 10.1007/s11139-017-9886-5 – volume: 144 start-page: 249 year: 1914 ident: ref_1 article-title: Neue Anwendungen der Theorie der Diophantischen Approximationen auf die Riemannsche Zetafunktion publication-title: J. Reine Angew. Math. doi: 10.1515/crll.1914.144.249 – ident: ref_4 – ident: ref_3 – volume: 25 start-page: 71 year: 2020 ident: ref_14 article-title: Joint discrete approximation of a pair of analytic functions by periodic zeta-functions publication-title: Math. Modell. Anal. doi: 10.3846/mma.2020.10450 – ident: ref_5 doi: 10.1007/978-94-017-2091-5 – volume: 204 start-page: 279 year: 2019 ident: ref_16 article-title: Universality of the Riemann zeta-function in short intervals publication-title: J. Number Theory doi: 10.1016/j.jnt.2019.04.006 – volume: 68 start-page: 741 year: 2018 ident: ref_11 article-title: The Riemann hypothesis and universality of the Riemann zeta-function publication-title: Math. Slovaca doi: 10.1515/ms-2017-0141 – volume: 108 start-page: 271 year: 2017 ident: ref_9 article-title: The discrete universality of the Riemann zeta-function with respect to uniformly distributed shifts publication-title: Arch. Math. doi: 10.1007/s00013-016-0998-8 – volume: 181 start-page: 127 year: 2017 ident: ref_12 article-title: Zeros of the Riemann zeta-function and its universality publication-title: Acta Arith. doi: 10.4064/aa8583-5-2017 – ident: ref_22 doi: 10.1007/BFb0060851 – volume: 34 start-page: 440 year: 1980 ident: ref_7 article-title: Werteverteilung von Zetafunktionen publication-title: Arch. Math. doi: 10.1007/BF01224983 – volume: 9 start-page: 443 year: 1975 ident: ref_2 article-title: Theorem on the “universality” of the Riemann zeta-function publication-title: Izv. Ross. Akad. Nauk. – ident: ref_17 – ident: ref_18 doi: 10.1515/9783110886146 – ident: ref_19 – ident: ref_13 doi: 10.1090/pspum/024 – volume: 210 start-page: 1753 year: 2019 ident: ref_15 article-title: Universality of Dirichlet L-functions and non-trivial zeros of the Riemann zeta-function publication-title: Sb. Math. doi: 10.1070/SM9194 – ident: ref_23 – ident: ref_21 – ident: ref_20 – volume: 30 start-page: 103 year: 2019 ident: ref_8 article-title: Discrete universality of the Riemann zeta-function and uniform distribution modulo 1 publication-title: Petersb. Math. J. |
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| Snippet | Let 0<γ1<γ2<⋯⩽γk⩽⋯ be the sequence of imaginary parts of non-trivial zeros of the Riemann zeta-function ζ(s). Using a certain estimate on the pair correlation... Let 0<γ1<γ2<⋯⩽γk⩽⋯ be the sequence of imaginary parts of non-trivial zeros of the Riemann zeta-function ζ ( s ) . Using a certain estimate on the pair... |
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| StartPage | 1936 |
| SubjectTerms | Montgomery pair correlation conjecture non-trivial zeros Riemann zeta-function universality |
| Title | Universality in Short Intervals of the Riemann Zeta-Function Twisted by Non-Trivial Zeros |
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