A central limit theorem for coefficients of the modified Borwein method for the calculation of the Riemann zeta-function

2019 ◽  
Vol 59 (1) ◽  
pp. 17-23 ◽  
Author(s):  
Igoris Belovas
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Zeta Function ◽  
Central Limit ◽  
Riemann Zeta Function ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

scholarly journals Bootstrapped zero density estimates and a central limit theorem for the zeros of the zeta function

2015 ◽  
Vol 11 (07) ◽  
pp. 2087-2107 ◽  
Author(s):  
Kenneth Maples ◽  
Brad Rodgers
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Zeta Function ◽  
Central Limit ◽  
Random Matrix ◽  
Matrix Theory ◽  
Density Estimates ◽  
Zero Density ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

We unconditionally prove a central limit theorem for linear statistics of the zeros of the Riemann zeta function with diverging variance. Previously, theorems of this sort have been proved under the assumption of the Riemann hypothesis. The result mirrors central limit theorems in random matrix theory that have been proved by Szegő, Spohn, and Soshnikov among others, and therefore provides support for the view that the zeros of the zeta function are distributed like the eigenvalues of a random matrix. A key ingredient in our proof is a simple bootstrapping of classical zero density estimates of Selberg and Jutila for the zeta function, which may be of independent interest.

scholarly journals Local limit theorem for coefficients of modified Borwein’s algorithm, proved by the ratio method

2019 ◽  
Vol 60 ◽  
pp. 11-14
Author(s):  
Igoris Belovas
Keyword(s):  
Limit Theorem ◽  
Zeta Function ◽  
Riemann Zeta Function ◽  
Local Limit Theorem ◽  
Local Limit ◽  
Ratio Method ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

The paper continues the research of the modified Borwein method for the evaluation of the Riemann zeta-function. It provides a different perspective on the derivation of the local limit theorem for coefficients of the method. The approach is based on the ratio method, proposed by Proschan.  

scholarly journals Weighted value distributions of the Riemann zeta function on the critical line

2021 ◽  
Vol 33 (3) ◽  
pp. 579-592
Author(s):  
Alessandro Fazzari
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Asymptotic Formula ◽  
Random Matrix Theory ◽  
Random Matrix ◽  
Riemann Zeta Function ◽  
Critical Line ◽  
Matrix Theory ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

Abstract We prove a central limit theorem for log ⁡ | ζ ⁢ ( 1 2 + i ⁢ t ) | {\log\lvert\zeta(\frac{1}{2}+it)\rvert} with respect to the measure | ζ ( m ) ⁢ ( 1 2 + i ⁢ t ) | 2 ⁢ k ⁢ d ⁢ t {\lvert\zeta^{(m)}(\frac{1}{2}+it)\rvert^{2k}\,dt} ( k , m ∈ ℕ {k,m\in\mathbb{N}} ), assuming RH and the asymptotic formula for twisted and shifted integral moments of zeta. Under the same hypotheses, we also study a shifted case, looking at the measure | ζ ⁢ ( 1 2 + i ⁢ t + i ⁢ α ) | 2 ⁢ k ⁢ d ⁢ t {\lvert\zeta(\frac{1}{2}+it+i\alpha)\rvert^{2k}\,dt} , with α ∈ ( - 1 , 1 ) {\alpha\in(-1,1)} . Finally, we prove unconditionally the analogue result in the random matrix theory context.

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