scholarly journals EULER PRODUCT ASYMPTOTICS FOR DIRICHLET -FUNCTIONS

2022 ◽  
pp. 1-9
Author(s):  
IKUYA KANEKO
Keyword(s):  
Riemann Hypothesis ◽  
Critical Line ◽  
Euler Product ◽  
Euler Products

Abstract The aim of this article is to establish the behaviour of partial Euler products for Dirichlet L-functions under the generalised Riemann hypothesis (GRH) via Ramanujan’s work. To understand the behaviour of Euler products on the critical line, we invoke the deep Riemann hypothesis (DRH). This work clarifies the relation between GRH and DRH.

Partial Euler Products on the Critical Line

2005 ◽  
Vol 57 (2) ◽  
pp. 267-297 ◽  
Author(s):  
Keith Conrad
Keyword(s):  
Asymptotic Behavior ◽  
Elliptic Curve ◽  
Riemann Hypothesis ◽  
Critical Line ◽  
Function Fields ◽  
Number Fields ◽  
Euler Product ◽  
Second Moments ◽  
Precise Sense ◽  
Euler Products

AbstractThe initial version of the Birch and Swinnerton-Dyer conjecture concerned asymptotics for partial Euler products for an elliptic curve L-function at s = 1. Goldfeld later proved that these asymptotics imply the Riemann hypothesis for the L-function and that the constant in the asymptotics has an unexpected factor of. We extend Goldfeld's theorem to an analysis of partial Euler products for a typical L-function along its critical line. The general phenomenon is related to second moments, while the asymptotic behavior (over number fields) is proved to be equivalent to a condition that in a precise sense seemsmuch deeper than the Riemann hypothesis. Over function fields, the Euler product asymptotics can sometimes be proved unconditionally.

scholarly journals An Analytical and Numerical Detour for the Riemann Hypothesis

Information ◽  
2021 ◽  
Vol 12 (11) ◽  
pp. 483
Author(s):  
Michel Riguidel
Keyword(s):  
Riemann Hypothesis ◽  
Meromorphic Functions ◽  
Critical Line ◽  
Numerical Approach ◽  
Product Formula ◽  
Critical Strip ◽  
Associated Functions ◽  
Poles And Zeros ◽  
Riemann’S Zeta Function ◽  
Insight Into

From the functional equation of Riemann’s zeta function, this article gives new insight into Hadamard’s product formula. The function and its family of associated functions, expressed as a sum of rational fractions, are interpreted as meromorphic functions whose poles are the poles and zeros of the function. This family is a mathematical and numerical tool which makes it possible to estimate the value of the function at a point in the critical strip from a point on the critical line .Generating estimates of at a given point requires a large number of adjacent zeros, due to the slow convergence of the series. The process allows a numerical approach of the Riemann hypothesis (RH). The method can be extended to other meromorphic functions, in the neighborhood of isolated zeros, inspired by the Weierstraß canonical form. A final and brief comparison is made with the and functions over finite fields.

On the Non-Existence of Certain Euler Products

1980 ◽  
Vol 23 (3) ◽  
pp. 371-372
Author(s):  
M. V. Subbarao
Keyword(s):  
Euler Product ◽  
Product Decomposition ◽  
Euler Products ◽  
Image Position ◽  
The Right

In a paper with the above title, T. M. Apostol and S. Chowla [1] proved the following result:Theorem 1.For relatively prime integers h and k, l ≤ h ≤ k, the seriesdoes not admit of an Euler product decomposition, that is, an identity of the form1except when h = k = l; fc = 1, fc = 2. The series on the right is extended over all primes p and is assumed to be absolutely convergent forR(s)>1.

scholarly journals The Values of the Periodic Zeta-Function at the Nontrivial Zeros of Riemann’s Zeta-Function

Symmetry ◽  
2021 ◽  
Vol 13 (12) ◽  
pp. 2410
Author(s):  
Janyarak Tongsomporn ◽  
Saeree Wananiyakul ◽  
Jörn Steuding
Keyword(s):  
Asymptotic Formula ◽  
Zeta Function ◽  
Riemann Zeta Function ◽  
Real Line ◽  
Riemann Hypothesis ◽  
Critical Line ◽  
Nontrivial Zeros ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Riemann’S Zeta Function

In this paper, we prove an asymptotic formula for the sum of the values of the periodic zeta-function at the nontrivial zeros of the Riemann zeta-function (up to some height) which are symmetrical on the real line and the critical line. This is an extension of the previous results due to Garunkštis, Kalpokas, and, more recently, Sowa. Whereas Sowa’s approach was assuming the yet unproved Riemann hypothesis, our result holds unconditionally.

scholarly journals Wigner Distribution Analysis and the Riemann Hypothesis

2020 ◽  
Vol 19 ◽  
Keyword(s):  
Signal Processing ◽  
Dirichlet Series ◽  
Riemann Hypothesis ◽  
Wigner Distribution ◽  
Möbius Function ◽  
Distribution Analysis ◽  
Euler Product ◽  
Instantaneous Spectrum ◽  
Mobius Function

Wigner distribution is a tool for signal processing to obtain instantaneous spectrum of a signal. From which, another representation of the Euler product can be obtained for Dirichlet series of the Mobius function, which leads to the proof of the Riemann hypothesis

scholarly journals Non-vanishing of Dirichlet series without Euler products

2018 ◽  
Vol Volume 40 ◽  
Author(s):  
William D. Banks
Keyword(s):  
Zeta Function ◽  
Dirichlet Series ◽  
Half Plane ◽  
Riemann Zeta Function ◽  
Euler Product ◽  
The Riemann Zeta Function ◽  
International Audience ◽  
Euler Products ◽  
Riemann Zeta

International audience We give a new proof that the Riemann zeta function is nonzero in the half-plane {s ∈ C : σ > 1}. A novel feature of this proof is that it makes no use of the Euler product for ζ(s).

scholarly journals On Ramanujan and Dirichlet series with Euler products

1984 ◽  
Vol 25 (2) ◽  
pp. 203-206 ◽  
Author(s):  
S. Raghavan
Keyword(s):  
Modular Form ◽  
Dirichlet Series ◽  
Modular Forms ◽  
Hecke Operators ◽  
Euler Product ◽  
Arithmetical Functions ◽  
In Trans ◽  
Euler Products ◽  
Unpublished Manuscripts ◽  
Profound Insight

In his unpublished manuscripts (referred to by Birch [1] as Fragment V, pp. 247–249), Ramanujan [3] gave a whole list of assertions about various (transforms of) modular forms possessing naturally associated Euler products, in more or less the spirit of his extremely beautiful paper entitled “On certain arithmetical functions” (in Trans. Camb. Phil. Soc. 22 (1916)). It is simply amazing how Ramanujan could write down (with an ostensibly profound insight) a basis of eigenfunctions (of Hecke operators) whose associated Dirichlet series have Euler products, anticipating by two decades the famous work of Hecke and Petersson. That he had further realized, in the event of a modular form f not corresponding to an Euler product, the possibility of restoring the Euler product property to a suitable linear combination of modular forms of the same type as f, is evidently fantastic.

scholarly journals The average multiplicative order of a finitely generated subgroup of the rationals modulo primes

2016 ◽  
Vol 12 (08) ◽  
pp. 2147-2158
Author(s):  
Cihan Pehlivan
Keyword(s):  
Prime Number ◽  
Explicit Expression ◽  
Riemann Hypothesis ◽  
Multiplicative Group ◽  
Prime Numbers ◽  
Euler Product ◽  
Finitely Generated ◽  
Numerical Computations ◽  
Multiplicative Order ◽  
Reduction Group

Let [Formula: see text] be a finitely generated subgroup of [Formula: see text]. Let [Formula: see text] be a prime number for which the reduction group [Formula: see text] is a well-defined subgroup of the multiplicative group [Formula: see text], and denote the order of [Formula: see text] by [Formula: see text]. Assuming the Generalized Riemann Hypothesis, we study the average of [Formula: see text] and the average of the powers of [Formula: see text] as [Formula: see text] ranges over prime numbers. The problem was considered in the case of rank 1 by Pomerance and Kurlberg. In the case when [Formula: see text] contains only positive numbers, we give an explicit expression for the involved density in terms of an Euler product. We conclude with some numerical computations.

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