Towards Euler’s Product Formula and Riemann’s Extension of the Zeta Function

2021 ◽  
pp. 1-6
Author(s):  
Walter Dittrich
Keyword(s):  
Zeta Function ◽  
Product Formula

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’s zeta function. This is a reciprocal relationship. The zeros and the primes are encoded in each other and are reciprocally recoverable. Riemann’s zeta is an extended or continued version of Euler’s zeta function which in turn equates with Euler’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.

ON THE EULER PRODUCT OF THE DEDEKIND ZETA FUNCTION

2009 ◽  
Vol 05 (02) ◽  
pp. 293-301
Author(s):  
XIAN-JIN LI
Keyword(s):  
Zeta Function ◽  
Algebraic Number ◽  
Zeta Functions ◽  
Algebraic Number Field ◽  
Number Fields ◽  
Product Formula ◽  
Euler Product ◽  
Dedekind Zeta Function ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

It is well known that the Euler product formula for the Riemann zeta function ζ(s) is still valid for ℜ(s) = 1 and s ≠ 1. In this paper, we extend this result to zeta functions of number fields. In particular, we show that the Dedekind zeta function ζk(s) for any algebraic number field k can be written as the Euler product on the line ℜ(s) = 1 except at the point s = 1. As a corollary, we obtain the Euler product formula on the line ℜ(s) = 1 for Dirichlet L-functions L(s, χ) of real characters.

Certain Subclasses of Bi-Univalent Functions Involving Double Zeta Function

2015 ◽  
Vol 4 (4) ◽  
pp. 28-33
Author(s):  
Dr. T. Ram Reddy ◽  
◽  
R. Bharavi Sharma ◽  
K. Rajya Lakshmi ◽  
◽  
...  
Keyword(s):  
Zeta Function ◽  
Univalent Functions ◽  
Double Zeta

Universality of $L$-Dirichlet functions and nontrivial zeros of the Riemann zeta-function

2019 ◽  
Vol 210 (12) ◽  
pp. 1753-1773 ◽  
Author(s):  
A. Laurinčikas ◽  
J. Petuškinaitė
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Nontrivial Zeros ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

The (α, β)-ramification invariants of a number field

2021 ◽  
Vol 71 (1) ◽  
pp. 251-263
Author(s):  
Guillermo Mantilla-Soler
Keyword(s):  
Zeta Function ◽  
Number Field ◽  
Residue Class ◽  
Interesting Application ◽  
Dedekind Zeta Function ◽  
Arithmetic Properties

Abstract Let L be a number field. For a given prime p, we define integers α p L $ \alpha_{p}^{L} $ and β p L $ \beta_{p}^{L} $ with some interesting arithmetic properties. For instance, β p L $ \beta_{p}^{L} $ is equal to 1 whenever p does not ramify in L and α p L $ \alpha_{p}^{L} $ is divisible by p whenever p is wildly ramified in L. The aforementioned properties, although interesting, follow easily from definitions; however a more interesting application of these invariants is the fact that they completely characterize the Dedekind zeta function of L. Moreover, if the residue class mod p of α p L $ \alpha_{p}^{L} $ is not zero for all p then such residues determine the genus of the integral trace.

scholarly journals Note on a Product Formula Related to Quantum Zeno Dynamics

2021 ◽  
Author(s):  
Pavel Exner ◽  
Takashi Ichinose
Keyword(s):  
Product Formula ◽  
Quantum Zeno Dynamics
Sign in / Sign up

Export Citation Format

Share Document