scholarly journals Pair correlation of the zeros of the Riemann zeta function in longer ranges

2004 ◽  
Vol 115 (2) ◽  
pp. 181-204 ◽  
Author(s):  
Tsz Ho Chan
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Pair Correlation ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

scholarly journals HIGHER CORRELATIONS AND THE ALTERNATIVE HYPOTHESIS

2020 ◽  
Vol 71 (1) ◽  
pp. 257-280
Author(s):  
Jeffrey C Lagarias ◽  
Brad Rodgers
Keyword(s):  
Correlation Function ◽  
Zeta Function ◽  
Point Process ◽  
Riemann Zeta Function ◽  
Alternative Hypothesis ◽  
Pair Correlation ◽  
Local Statistics ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Rule Out

Abstract The Alternative Hypothesis (AH) concerns a hypothetical and unlikely picture of how zeros of the Riemann zeta function are spaced, which one would like to rule out. In the Alternative Hypothesis, the renormalized distance between non-trivial zeros is supposed to always lie at a half integer. It is known that the Alternative Hypothesis is compatible with what is known about the pair correlation function of zeta zeros. We ask whether what is currently known about higher correlation functions of the zeros is sufficient to rule out the Alternative Hypothesis and show by construction of an explicit counterexample point process that it is not. A similar result was recently independently obtained by Tao, using slightly different methods. We also apply the ergodic theorem to this point process to show there exists a deterministic collection of points lying in $\tfrac{1}{2}\mathbb{Z}$, which satisfy the Alternative Hypothesis spacing, but mimic the local statistics that are currently known about zeros of the zeta function.

On the Pair Correlation of Zeros of the Riemann Zeta-Function

2000 ◽  
Vol 80 (1) ◽  
pp. 31-49 ◽  
Author(s):  
D. A. Goldston ◽  
S. M. Gonek ◽  
A. E. Özlük ◽  
C. Snyder
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Pair Correlation ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

scholarly journals UNIVERSALITY OF ZETA-FUNCTIONS OF CUSP FORMS AND NON-TRIVIAL ZEROS OF THE RIEMANN ZETA-FUNCTION

2021 ◽  
Vol 26 (1) ◽  
pp. 82-93
Author(s):  
Aidas Balčiūnas ◽  
Violeta Franckevič ◽  
Virginija Garbaliauskienė ◽  
Renata Macaitienė ◽  
Audronė Rimkevičienė
Keyword(s):  
Zeta Function ◽  
Wide Class ◽  
Riemann Zeta Function ◽  
Analytic Functions ◽  
Pair Correlation ◽  
Zeta Functions ◽  
Weak Form ◽  
Cusp Forms ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

It is known that zeta-functions ζ(s,F) of normalized Hecke-eigen cusp forms F are universal in the Voronin sense, i.e., their shifts ζ(s + iτ,F), τ R, approximate a wide class of analytic functions. In the paper, under a weak form of the Montgomery pair correlation conjecture, it is proved that the shifts ζ(s+iγkh,F), where γ1 < γ2 < ... is a sequence of imaginary parts of non-trivial zeros of the Riemann zeta function and h > 0, also approximate a wide class of analytic functions.

scholarly journals Universality in Short Intervals of the Riemann Zeta-Function Twisted by Non-Trivial Zeros

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 1936
Author(s):  
Antanas Laurinčikas ◽  
Darius Šiaučiūnas
Keyword(s):  
Analytic Function ◽  
Zeta Function ◽  
Riemann Zeta Function ◽  
Approximation Property ◽  
Pair Correlation ◽  
Positive Density ◽  
Short Intervals ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

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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