scholarly journals Selberg's zeta function for the modular group in the critical strip

2021 ◽  
Author(s):  
Yasufumi Hashimoto
Keyword(s):  
Zeta Function ◽  
Modular Group ◽  
Critical Strip ◽  
Selberg’S Zeta Function

scholarly journals On the integer part of the reciprocal of the Riemann zeta function tail at certain rational numbers in the critical strip

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
WonTae Hwang ◽  
Kyunghwan Song
Keyword(s):  
Zeta Function ◽  
Rational Number ◽  
Riemann Zeta Function ◽  
Rational Numbers ◽  
Critical Strip ◽  
Integer Part ◽  
Integral Points ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

Abstract We prove that the integer part of the reciprocal of the tail of $\zeta (s)$ ζ ( s ) at a rational number $s=\frac{1}{p}$ s = 1 p for any integer with $p \geq 5$ p ≥ 5 or $s=\frac{2}{p}$ s = 2 p for any odd integer with $p \geq 5$ p ≥ 5 can be described essentially as the integer part of an explicit quantity corresponding to it. To deal with the case when $s=\frac{2}{p}$ s = 2 p , we use a result on the finiteness of integral points of certain curves over $\mathbb{Q}$ Q .

scholarly journals Selberg's zeta function and the spectral geometry of geometrically finite hyperbolic surfaces

2005 ◽  
pp. 483-515 ◽  
Author(s):  
David Borthwick ◽  
Chris Judge ◽  
Peter Perry
Keyword(s):  
Zeta Function ◽  
Spectral Geometry ◽  
Hyperbolic Surfaces ◽  
Geometrically Finite ◽  
Selberg’S Zeta Function

scholarly journals Morphogenesis of the Zeta Function in the Critical Strip by Computational Approach

Mathematics ◽  
2018 ◽  
Vol 6 (12) ◽  
pp. 285
Author(s):  
Michel Riguidel
Keyword(s):  
Zeta Function ◽  
Numerical Approximation ◽  
Critical Line ◽  
Formal Proof ◽  
Computational Approach ◽  
Partial Sums ◽  
Critical Strip ◽  
Power Functions ◽  
Elementary Functions ◽  
2D And 3D

This article proposes a morphogenesis interpretation of the zeta function by computational approach by relying on numerical approximation formulae between the terms and the partial sums of the series, divergent in the critical strip. The goal is to exhibit structuring properties of the partial sums of the raw series by highlighting their morphogenesis, thanks to the elementary functions constituting the terms of the real and imaginary parts of the series, namely the logarithmic, cosine, sine, and power functions. Two essential indices of these sums appear: the index of no return of the vagrancy and the index of smothering of the function before the resumption of amplification of its divergence when the index tends towards infinity. The method consists of calculating, displaying graphically in 2D and 3D, and correlating, according to the index, the angles, the terms and the partial sums, in three nested domains: the critical strip, the critical line, and the set of non-trivial zeros on this line. Characteristics and approximation formulae are thus identified for the three domains. These formulae make it possible to grasp the morphogenetic foundations of the Riemann hypothesis (RH) and sketch the architecture of a more formal proof.

scholarly journals ON THE MODULUS OF THE SELBERG ZETA-FUNCTIONS IN THE CRITICAL STRIP

2015 ◽  
Vol 20 (6) ◽  
pp. 852-865
Author(s):  
Andrius Grigutis ◽  
Darius Šiaučiūnas
Keyword(s):  
Riemann Surface ◽  
Modular Group ◽  
Compact Riemann Surface ◽  
Zeta Functions ◽  
Critical Strip ◽  
The Real ◽  
Derivatives Of ◽  
Logarithmic Derivatives

We investigate the behavior of the real part of the logarithmic derivatives of the Selberg zeta-functions ZPSL(2,Z)(s) and ZC (s) in the critical strip 0 < σ < 1. The functions ZPSL(2,Z)(s) and ZC (s) are defined on the modular group and on the compact Riemann surface, respectively.

Mayer’s transfer operator approach to Selberg’s zeta function

2013 ◽  
Vol 24 (4) ◽  
pp. 529-553
Author(s):  
A. Momeni ◽  
A. B. Venkov
Keyword(s):  
Zeta Function ◽  
Transfer Operator ◽  
Operator Approach ◽  
Selberg’S Zeta Function

scholarly journals The Zeros of the Dirichlet Beta Function Encode the Odd Primes and Have Real Part 1/2

2018 ◽  
Author(s):  
Anthony Lander
Keyword(s):  
Zeta Function ◽  
Beta Function ◽  
Product Formula ◽  
Reciprocal Relationship ◽  
Critical Strip ◽  
The Real ◽  
Nontrivial Zeros ◽  
Deep Relationship

It is well known that the primes and prime powers have a deep relationship with the nontrivial zeros of Riemann&rsquo;s zeta function. This is a reciprocal relationship. The zeros and the primes are encoded in each other and are reciprocally recoverable. Riemann&rsquo;s zeta is an extended or continued version of Euler&rsquo;s zeta function which in turn equates with Euler&rsquo;s product formula over the primes. This paper shows that the zeros of the converging Dirichlet or Catalan beta function, which requires no continuation to be valid in the critical strip, can be easily determined. The imaginary parts of these zeros have a deep and reciprocal relationship with the odd primes and odd prime powers. This relationship separates the odd primes into those having either 1 or 3 as a remainder after division by 4. The vector pathway of the beta function is such that the real part of its zeros has to be a half.

scholarly journals On the zeros of the Riemann zeta function in the critical strip. III

1983 ◽  
Vol 41 (164) ◽  
pp. 759-759 ◽  
Author(s):  
J. van de Lune ◽  
H. J. J. te Riele
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Critical Strip ◽  
The Riemann Zeta Function ◽  
Riemann Zeta
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