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Jensen polynomials for the Riemann zeta function and other sequences. [PDF]

Proc Natl Acad Sci U S A, 2019
In 1927 P\'olya proved that the Riemann Hypothesis is equivalent to the hyperbolicity of Jensen polynomials for the Riemann zeta function $\zeta(s)$ at its point of symmetry. This hyperbolicity has been proved for degrees $d\leq 3$.
Griffin M, Ono K, Rolen L, Zagier D.
europepmc   +6 more sources

The inverses of tails of the Riemann zeta function [PDF]

Journal of Inequalities and Applications, 2018
We present some bounds of the inverses of tails of the Riemann zeta function on ...
Donggyun Kim, Kyunghwan Song
doaj   +4 more sources

q-Riemann zeta function [PDF]

International Journal of Mathematics and Mathematical Sciences, 2004
We consider the modified q-analogue of Riemann zeta function which is defined by ζq(s)=∑n=1∞(qn(s−1)/[n]s ...
Taekyun Kim
doaj   +2 more sources

Scalar modular bootstrap and zeros of the Riemann zeta function [PDF]

Journal of High Energy Physics, 2022
Using the technology of harmonic analysis, we derive a crossing equation that acts only on the scalar primary operators of any two-dimensional conformal field theory with U(1) c symmetry. From this crossing equation, we derive bounds on the scalar gap of
Nathan Benjamin, Cyuan-Han Chang
doaj   +2 more sources

Explicit zero-free regions for the Riemann zeta-function [PDF]

Research in Number Theory, 2022
We prove that the Riemann zeta-function ζ(σ+it)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin ...
Michael J. Mossinghoff, T. Trudgian, Andrew Yang   +2 more
semanticscholar   +1 more source

Correlations of the Riemann zeta function [PDF]

Mathematika, 2023
Assuming the Riemann hypothesis, we investigate the shifted moments of the zeta function Mα,β(T)=∫T2T∏k=1mζ12+i(t+αk)2βkdt$$\begin{equation*} \hspace*{24.5pt}M_{{\bm \alpha}, {\bm \beta}} (T) = \int _T^{2T} \prod _{k = 1}^m {\left|\zeta \left(\tfrac{1}{2}
Michael J. Curran
semanticscholar   +1 more source

Counting zeros of the Riemann zeta function [PDF]

Journal of Number Theory, 2021
In this article, we show that | N ( T ) − T 2 π log ⁡ ( T 2 π e ) | ≤ 0.1038 log ⁡ T + 0.2573 log ⁡ log ⁡ T + 9.3675 where N ( T ) denotes the number of non-trivial zeros ρ, with 0 Im ( ρ ) ≤ T , of the Riemann zeta function.
Elchin Hasanalizade, Quanli Shen, PENG-JIE Wong   +2 more
semanticscholar   +1 more source

Extended Riemann Zeta Functions [PDF]

Rocky Mountain Journal of Mathematics, 2001
In this paper two extensions of the Riemann zeta-function are presented. Denoting the real part of the complex variable \(s\) by \(\sigma\), these extensions are \[ \zeta_{b}(s) = {1\over \Gamma(s)}\int_{0}^{\infty}t^{s-1}(e^{t}-1)^{-1}e^{-b/t} dt \quad \quad (b>0; b=0, \sigma >1), \] and \[ \zeta_{b}^{*}(s) = {1\over \Gamma(s)(1-2^{1-s})}\int_{0 ...
Chaudhry, M. Aslam, Qadir, Asghar, Boudjelkha, M.T., Rafique, M., Zubair, S.M.   +4 more
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Shifted moments of the Riemann zeta function [PDF]

Canadian Journal of Mathematics - Journal Canadien de Mathematiques, 2022
In this article, we prove that the Riemann hypothesis implies a conjecture of Chandee on shifted moments of the Riemann zeta function. The proof is based on ideas of Harper concerning sharp upper bounds for the $2k$ th moments of the Riemann zeta ...
Nathan Ng, Quanli Shen, PENG-JIE Wong
semanticscholar   +1 more source

Amplitudes and the Riemann Zeta Function. [PDF]

Physical Review Letters, 2021
Physical properties of scattering amplitudes are mapped to the Riemann zeta function. Specifically, a closed-form amplitude is constructed, describing the tree-level exchange of a tower with masses m_{n}^{2}=μ_{n}^{2}, where ζ(1/2±iμ_{n})=0.
Grant N. Remmen
semanticscholar   +1 more source