scholarly journals On a Li-type criterion for zero-free regions of certain Dirichlet series with real coefficients

2016 ◽  
Vol 19 (1) ◽  
pp. 259-280 ◽  
Author(s):  
Alina Bucur ◽  
Anne-Maria Ernvall-Hytönen ◽  
Almasa Odžak ◽  
Lejla Smajlović
Keyword(s):  
Zeta Function ◽  
Dirichlet Series ◽  
Half Plane ◽  
Riemann Hypothesis ◽  
Selberg Class ◽  
Generalized Riemann Hypothesis ◽  
Type Criterion ◽  
Extended Selberg Class

The Li coefficients $\unicode[STIX]{x1D706}_{F}(n)$ of a zeta or $L$-function $F$ provide an equivalent criterion for the (generalized) Riemann hypothesis. In this paper we define these coefficients, and their generalizations, the $\unicode[STIX]{x1D70F}$-Li coefficients, for a subclass of the extended Selberg class which is known to contain functions violating the Riemann hypothesis such as the Davenport–Heilbronn zeta function. The behavior of the $\unicode[STIX]{x1D70F}$-Li coefficients varies depending on whether the function in question has any zeros in the half-plane $\text{Re}(z)>\unicode[STIX]{x1D70F}/2.$ We investigate analytically and numerically the behavior of these coefficients for such functions in both the $n$ and $\unicode[STIX]{x1D70F}$ aspects.

scholarly journals Variant of the truncated Perron formula and primes in polynomial sets

2019 ◽  
Vol 16 (02) ◽  
pp. 309-323
Author(s):  
D. S. Ramana ◽  
O. Ramaré
Keyword(s):  
Dirichlet Series ◽  
Riemann Hypothesis ◽  
Partial Sums ◽  
Generalized Riemann Hypothesis ◽  
Perron’S Formula ◽  
Polynomial Formula

We show under the Generalized Riemann Hypothesis that for every non-constant integer-valued polynomial [Formula: see text], for every [Formula: see text], and almost every prime [Formula: see text] in [Formula: see text], the number of primes from the interval [Formula: see text] that are values of [Formula: see text] modulo [Formula: see text] is the expected one, provided [Formula: see text] is not more than [Formula: see text]. We obtain this via a variant of the classical truncated Perron’s formula for the partial sums of the coefficients of a Dirichlet series.

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 UNE MINORATION POUR L'EXPOSANT DU GROUPE DES CLASSES D'UN CORPS ENGENDRÉ PAR UN NOMBRE DE SALEM

2007 ◽  
Vol 03 (02) ◽  
pp. 217-229 ◽  
Author(s):  
FRANCESCO AMOROSO
Keyword(s):  
Zeta Function ◽  
Number Field ◽  
Lower Bounds ◽  
Riemann Hypothesis ◽  
Class Group ◽  
Relative Size ◽  
Generalized Riemann Hypothesis ◽  
Dedekind Zeta Function ◽  
Salem Number

In this article we extend the main result of [2] concerning lower bounds for the exponent of the class group of CM-fields. We consider a number field K generated by a Salem number α. If k denotes the field fixed by α ↦ α-1 we prove, under the generalized Riemann hypothesis for the Dedekind zeta function of K, lower bounds for the relative exponent eK/k and the relative size hK/k of the class group of K with respect to the class group of k.

scholarly journals The Saddle-Point Method and the Li Coefficients

2011 ◽  
Vol 54 (2) ◽  
pp. 316-329
Author(s):  
Kamel Mazhouda
Keyword(s):  
Asymptotic Formula ◽  
Saddle Point ◽  
Riemann Hypothesis ◽  
Point Method ◽  
Saddle Point Method ◽  
Selberg Class ◽  
Generalized Riemann Hypothesis ◽  
Image Position ◽  
Euler's Constant ◽  
Precise Asymptotic

AbstractIn this paper, we apply the saddle-point method in conjunction with the theory of the Nörlund–Rice integrals to derive precise asymptotic formula for the generalized Li coefficients established by Omar and Mazhouda. Actually, for any function F in the Selberg class and under the Generalized Riemann Hypothesis, we havewithwhere γ is the Euler's constant and the notation is as below.

scholarly journals Stieltjes constants of L-functions in the extended Selberg class

2021 ◽  
Author(s):  
Shōta Inoue ◽  
Sumaia Saad Eddin ◽  
Ade Irma Suriajaya
Keyword(s):  
Dirichlet Series ◽  
Upper Bound ◽  
Arithmetic Function ◽  
Laurent Expansion ◽  
Selberg Class ◽  
Stieltjes Constants ◽  
Extended Selberg Class

AbstractLet f be an arithmetic function and let $${\mathcal {S}}^\#$$ S # denote the extended Selberg class. We denote by $${\mathcal {L}}(s) = \sum _{n = 1}^{\infty }\frac{f(n)}{n^s}$$ L ( s ) = ∑ n = 1 ∞ f ( n ) n s the Dirichlet series attached to f. The Laurent–Stieltjes constants of $${\mathcal {L}}(s)$$ L ( s ) , which belongs to $${\mathcal {S}}^\#$$ S # , are the coefficients of the Laurent expansion of $${\mathcal {L}}$$ L at its pole $$s=1$$ s = 1 . In this paper, we give an upper bound of these constants, which is a generalization of many known results.

scholarly journals The Jensen–Pólya program for various L-functions

2020 ◽  
Vol 32 (2) ◽  
pp. 525-539
Author(s):  
Ian Wagner
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Riemann Hypothesis ◽  
Generalized Riemann Hypothesis ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

AbstractPólya proved in 1927 that the Riemann hypothesis is equivalent to the hyperbolicity of all of the Jensen polynomials of degree d and shift n for the Riemann Xi-function. Recently, Griffin, Ono, Rolen, and Zagier [M. Griffin, K. Ono, L. Rolen and D. Zagier, Jensen polynomials for the Riemann zeta function and other sequences, Proc. Natl. Acad. Sci. USA 116 2019, 23, 11103–11110] proved that for each degree {d\geq 1} all of the Jensen polynomials for the Riemann Xi-function are hyperbolic except for possibly finitely many n. Here we extend their work by showing that the same statement is true for suitable L-functions. This offers evidence for the generalized Riemann hypothesis.

UNIQUENESS THEOREMS FOR DIRICHLET SERIES

2015 ◽  
Vol 91 (3) ◽  
pp. 389-399 ◽  
Author(s):  
AI-DI WU ◽  
PEI-CHU HU
Keyword(s):  
Dirichlet Series ◽  
Uniqueness Theorems ◽  
Selberg Class ◽  
Finite Set ◽  
Extended Selberg Class

We obtain uniqueness theorems for L-functions in the extended Selberg class when the functions share values in a finite set and share values weighted by multiplicities.

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