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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Bibliographic Details
Published in:	Mathematics (Basel) Vol. 8; no. 11; p. 1936
Main Authors:	Laurinčikas, Antanas, Šiaučiūnas, Darius
Format:	Journal Article
Language:	English
Published:	MDPI AG 01.11.2020
Subjects:	Montgomery pair correlation conjecture non-trivial zeros Riemann zeta-function universality
ISSN:	2227-7390, 2227-7390
Online Access:	Get full text
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Summary:	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)).
ISSN:	2227-7390 2227-7390
DOI:	10.3390/math8111936