ZERO-FREE REGIONS OF A q-ANALOGUE OF THE COMPLETE RIEMANN ZETA FUNCTION

2011 ◽  
Vol 07 (04) ◽  
pp. 1075-1092
Author(s):  
MITSUGU MERA
Keyword(s):  
Zeta Function ◽  
Theta Function ◽  
Riemann Zeta Function ◽  
Mellin Transform ◽  
Critical Strip ◽  
Jacobi Theta Function ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

A q-analogue of the complete Riemann zeta function presented in this paper is defined by the q-Mellin transform of the Jacobi theta function. We study zero-free regions of the q-zeta function. As a by-product, we show that the Riemann zeta function does not vanish in a sub-region of the critical strip.

scholarly journals Some Mellin transforms for the Riemann zeta function in the critical strip

2019 ◽  
Vol 15 (01) ◽  
pp. 153-156
Author(s):  
Alexander E Patkowski
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Mellin Transform ◽  
Critical Strip ◽  
Mellin Transforms ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Fourier Integrals ◽  
The Way

We offer two new Mellin transform evaluations for the Riemann zeta function in the region [Formula: see text]. Some discussion is offered in the way of evaluating some further Fourier integrals involving the Riemann xi function.

scholarly journals On the integer part of the reciprocal of the Riemann zeta function tail at certain rational numbers in the critical strip

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
WonTae Hwang ◽  
Kyunghwan Song
Keyword(s):  
Zeta Function ◽  
Rational Number ◽  
Riemann Zeta Function ◽  
Rational Numbers ◽  
Critical Strip ◽  
Integer Part ◽  
Integral Points ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

Abstract We prove that the integer part of the reciprocal of the tail of $\zeta (s)$ ζ ( s ) at a rational number $s=\frac{1}{p}$ s = 1 p for any integer with $p \geq 5$ p ≥ 5 or $s=\frac{2}{p}$ s = 2 p for any odd integer with $p \geq 5$ p ≥ 5 can be described essentially as the integer part of an explicit quantity corresponding to it. To deal with the case when $s=\frac{2}{p}$ s = 2 p , we use a result on the finiteness of integral points of certain curves over $\mathbb{Q}$ Q .

scholarly journals SELF-APPROXIMATION FOR THE RIEMANN ZETA FUNCTION

2012 ◽  
Vol 87 (3) ◽  
pp. 452-461 ◽  
Author(s):  
TAKASHI NAKAMURA ◽  
ŁUKASZ PAŃKOWSKI
Keyword(s):  
Zeta Function ◽  
Half Plane ◽  
Riemann Zeta Function ◽  
Critical Strip ◽  
Approximation Of Functions ◽  
Joint Universality ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
The Right

AbstractIn the paper we deal with self-approximation of the Riemann zeta function in the half plane $\operatorname {Re} s\gt 1$ and in the right half of the critical strip. We also prove some results concerning joint universality and joint value approximation of functions $\zeta (s+\lambda +id\tau )$ and $\zeta (s+i\tau )$.

On large values of L(σ,χ)

2018 ◽  
Vol 70 (3) ◽  
pp. 831-848 ◽  
Author(s):  
Christoph Aistleitner ◽  
Kamalakshya Mahatab ◽  
Marc Munsch ◽  
Alexandre Peyrot
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Probabilistic Models ◽  
Extreme Values ◽  
Resonance Method ◽  
Critical Strip ◽  
Principal Character ◽  
Large Order ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

Abstract In recent years, a variant of the resonance method was developed which allowed to obtain improved Ω-results for the Riemann zeta function along vertical lines in the critical strip. In the present paper, we show how this method can be adapted to prove the existence of large values of |L(σ,χ)| in the range σ∈(1/2,1], and to estimate the proportion of characters for which |L(σ,χ)| is of such a large order. More precisely, for every fixed σ∈(1/2,1), we show that for all sufficiently large q, there is a non-principal character χ(modq) such that log|L(σ,χ)|≥C(σ)(logq)1−σ(loglogq)−σ. In the case σ=1, we show that there is a non-principal character χ(modq) for which |L(1,χ)|≥eγ(log2q+log3q−C). In both cases, our results essentially match the prediction for the actual order of such extreme values, based on probabilistic models.

scholarly journals Discrete Approximation by a Dirichlet Series Connected to the Riemann Zeta-Function

Mathematics ◽  
2021 ◽  
Vol 9 (10) ◽  
pp. 1073
Author(s):  
Antanas Laurinčikas ◽  
Darius Šiaučiūnas
Keyword(s):  
Zeta Function ◽  
Dirichlet Series ◽  
Riemann Zeta Function ◽  
Discrete Approximation ◽  
Critical Strip ◽  
Right Hand ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
The Right

In the paper, a Dirichlet series ζuN(s) whose shifts ζuN(s+ikh), k=0,1,⋯, h>0, approximate analytic non-vanishing functions defined on the right-hand side of the critical strip is considered. This series is closely connected to the Riemann zeta-function. The sequence uN→∞ and uN≪N2 as N→∞.

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