scholarly journals Ingham Tauberian theorem with an estimate for the error term

2003 ◽  
Vol 2003 (64) ◽  
pp. 4025-4031
Author(s):  
E. P. Balanzario ◽  
E. Marmolejo-Olea
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Tauberian Theorem ◽  
Error Term ◽  
Elementary Proof ◽  
Weak Form ◽  
Free Region ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

We estimate the error term in the Ingham Tauberian theorem. This estimation of the error term is accomplished by considering an elementary proof of a weak form of Wiener's general Tauberian theorem and by using a zero-free region for the Riemann zeta function.

scholarly journals UNIVERSALITY OF ZETA-FUNCTIONS OF CUSP FORMS AND NON-TRIVIAL ZEROS OF THE RIEMANN ZETA-FUNCTION

2021 ◽  
Vol 26 (1) ◽  
pp. 82-93
Author(s):  
Aidas Balčiūnas ◽  
Violeta Franckevič ◽  
Virginija Garbaliauskienė ◽  
Renata Macaitienė ◽  
Audronė Rimkevičienė
Keyword(s):  
Zeta Function ◽  
Wide Class ◽  
Riemann Zeta Function ◽  
Analytic Functions ◽  
Pair Correlation ◽  
Zeta Functions ◽  
Weak Form ◽  
Cusp Forms ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

It is known that zeta-functions ζ(s,F) of normalized Hecke-eigen cusp forms F are universal in the Voronin sense, i.e., their shifts ζ(s + iτ,F), τ R, approximate a wide class of analytic functions. In the paper, under a weak form of the Montgomery pair correlation conjecture, it is proved that the shifts ζ(s+iγkh,F), where γ1 < γ2 < ... is a sequence of imaginary parts of non-trivial zeros of the Riemann zeta function and h > 0, also approximate a wide class of analytic functions.

On the Riemann–Siegel formula

2007 ◽  
Vol 463 (2086) ◽  
pp. 2557-2568 ◽  
Author(s):  
A Kuznetsov
Keyword(s):  
Asymptotic Formula ◽  
Zeta Function ◽  
Critical Point ◽  
Riemann Zeta Function ◽  
Error Term ◽  
Error Function ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

In this article, we derive a generalization of the Riemann–Siegel asymptotic formula for the Riemann zeta function. By subtracting the singularities closest to the critical point, we obtain a significant reduction of the error term at the expense of a few evaluations of the error function. We illustrate the efficiency of this method by comparing it to the classical Riemann–Siegel formula.

Zeros of the Riemann zeta-function in the discrete universality of the Hurwitz zeta-function

2020 ◽  
Vol 57 (2) ◽  
pp. 147-164
Author(s):  
Antanas Laurinčikas
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Pair Correlation ◽  
Weak Form ◽  
Rational Numbers ◽  
Hurwitz Zeta Function ◽  
Linearly Independent ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Discrete Universality

AbstractLet 0 < γ1 < γ2 < ··· ⩽ ··· be the imaginary parts of non-trivial zeros of the Riemann zeta-function. In the paper, we consider the approximation of analytic functions by shifts of the Hurwitz zeta-function ζ(s + iγkh, α), h > 0, with parameter α such that the set {log(m + α): m ∈ } is linearly independent over the field of rational numbers. For this, a weak form of the Montgomery conjecture on the pair correlation of {γk} is applied.

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