Results 11 to 20 of about 106,424 (299)
Prime number theory and the Riemann zeta-function [PDF]
, 2005 D. R. Heath‐Brown
core +6 more sources
A note on prime zeta function and Riemann zeta function. Corrigendum [PDF]
Notes on Number Theory and Discrete Mathematics, 2021 In [1] the author proposed two new results concerning the prime zeta function and the Riemann zeta function but they turn out to be wrong. In the present paper we provide their correct form.
M. Vassilev-Missana
semanticscholar +2 more sources
The prime number theorem and pair correlation of zeros of the Riemann zeta-function [PDF]
Research in Number Theory, 2022 We prove that the error in the prime number theorem can be quantitatively improved beyond the Riemann Hypothesis bound by using versions of Montgomery’s conjecture for the pair correlation of zeros of the Riemann zeta-function which are uniform in long ...
D. Goldston, Ade Irma Suriajaya
semanticscholar +3 more sources
Along the Lines of Nonadditive Entropies: q-Prime Numbers and q-Zeta Functions [PDF]
Entropy, 2021 The rich history of prime numbers includes great names such as Euclid, who first analytically studied the prime numbers and proved that there is an infinite number of them, Euler, who introduced the function ζ(s)≡∑n=1∞n−s=∏pprime11−p−s, Gauss, who ...
Ernesto P. Borges +2 more
semanticscholar +6 more sources
A pseudo zeta function and the distribution of primes [PDF]
Proceedings of the National Academy of Sciences, 2000 The Riemann zeta function is given by: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{equation*}{\zeta}(s)={ \,\substack{ ^{{\infty}} \\ {\sum}
Paul R. Chernoff
openalex +5 more sources
Zeta function zeros, powers of primes, and quantum chaos [PDF]
Physical Review E, 2003 We present a numerical study of Riemann's formula for the oscillating part of the density of the primes and their powers. The formula is comprised of an infinite series of oscillatory terms, one for each zero of the zeta function on the critical line and was derived by Riemann in his paper on primes assuming the Riemann hypothesis.
Jamal Sakhr +2 more
openalex +6 more sources
On primeness of the Selberg zeta-function [PDF]
Hokkaido Mathematical Journal, 2020 Comment: To appear in Hokkaido Mathematical ...
Ramūnas Garunkštis, Jörn Steuding
openalex +5 more sources
On a prime zeta function of a graph [PDF]
Pacific Journal of Mathematics, 2015 Takehiro Hasegawa, S. Saito
semanticscholar +3 more sources
The pair correlation of zeros of the Riemann zeta function and distribution of primes [PDF]
Archiv der Mathematik, 2001 In 1972 \textit{H. L. Montgomery} [Analytic number theory, Proc. Sympos. Pure Math. 24, 181-193 (1973; Zbl 0268.10023)] introduced the function \[ G(T,\xi) = \sum_{0< \gamma_1 , \gamma_2 \leq T}e(\xi(\gamma_1-\gamma_2))w(\gamma_1-\gamma_2), \] where \(w(u)= 4/(4+u^2)\), \(e(u)= e^{2\pi iu}\), and the sum is over pairs of imaginary parts of zeros \(\rho=
Jingwen Liu, Y. Ye
openalex +2 more sources
Arithmetic forms of Selberg zeta functions with applications to prime geodesic theorem [PDF]
Proceedings of the Japan Academy, Series A, Mathematical Sciences, 2002 Excerpts from portions of the introduction: ``Let \(\Gamma\) be a discrete subgroup of \(\text{SL}_2(\mathbb{R})\) containing \(-1_2\) with finite covolume \(v(\Gamma\setminus{\mathfrak H})\), \({\mathfrak H}\) denoting the upper half plane. The Selberg zeta-function attached to \(\Gamma\) is defined by \[ Z_\Gamma(s):= \prod_{\{P\}_\Gamma} \prod_{m=0}^
Tsuneo Arakawa +2 more
openalex +4 more sources