scholarly journals Zeros of the Alternating Zeta Function on the Line R(S) = 1

2003 ◽  
Vol 110 (5) ◽  
pp. 435 ◽  
Author(s):  
Jonathan Sondow
Keyword(s):  
Zeta Function ◽  
Alternating Zeta Function

scholarly journals On the existence of equivalent Dirichlet polynomials whose zeros preserve a topological property

2018 ◽  
Vol 14 (03) ◽  
pp. 713-725
Author(s):  
Eric Dubon ◽  
Juan Matías Sepulcre
Keyword(s):  
Zeta Function ◽  
Equivalence Relation ◽  
Topological Property ◽  
Vertical Line ◽  
Partial Sums ◽  
Critical Strip ◽  
Multiplicative Functions ◽  
Distribution Of Zeros ◽  
Dirichlet Polynomials ◽  
Alternating Zeta Function

In this paper, we study the distribution of zeros of the ordinary Dirichlet polynomials which are generated by an equivalence relation introduced by Harald Bohr. Through the use of completely multiplicative functions, we construct equivalent Dirichlet polynomials which have the same critical strip, where all their zeros are situated, and satisfy the same topological property consisting of possessing zeros arbitrarily near every vertical line contained in some substrips inside their critical strip. We also show that the real projections of the zeros of the partial sums of the alternating zeta function, for some particular cases, are dense in their critical intervals.

scholarly journals Integral and Series Representations of Riemann's Zeta Function and Dirichlet's Eta Function and a Medley of Related Results

2013 ◽  
Vol 2013 ◽  
pp. 1-17 ◽  
Author(s):  
Michael S. Milgram
Keyword(s):  
Zeta Function ◽  
Integral Representations ◽  
Contour Integral ◽  
Series Representations ◽  
Riemann’S Zeta Function ◽  
Alternating Zeta Function

Contour integral representations of Riemann's Zeta function and Dirichlet's Eta (alternating Zeta) function are presented and investigated. These representations flow naturally from methods developed in the 1800s, but somehow they do not appear in the standard reference summaries, textbooks, or literature. Using these representations as a basis, alternate derivations of known series and integral representations for the Zeta and Eta function are obtained on a unified basis that differs from the textbook approach, and results are developed that appear to be new.

A weighted alternating zeta function of a digraph

2021 ◽  
pp. 1-23
Author(s):  
Takashi Komatsu ◽  
Norio Konno ◽  
Iwao Sato
Keyword(s):  
Zeta Function ◽  
Alternating Zeta Function

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

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.

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