scholarly journals Low Pseudomoments of Euler Products

2021 ◽  
Author(s):  
Maxim Gerspach ◽  
Youness Lamzouri
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
The Riemann Zeta Function ◽  
Order Of Magnitude ◽  
Euler Products ◽  
Riemann Zeta

Abstract In this paper, we determine the order of magnitude of the 2 q-th pseudomoment of powers of the Riemann zeta function $\zeta(s)^{\alpha}$ for $0 \lt q\le 1/2$ and $0 \lt \alpha \lt 1$, completing the results of Bondarenko, Heap and Seip, and Gerspach. Our results also apply to more general Euler products satisfying certain conditions.

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 Low Pseudomoments of the Riemann Zeta Function and Its Powers

2020 ◽  
Author(s):  
Maxim Gerspach
Keyword(s):  
Zeta Function ◽  
Lower Bounds ◽  
Riemann Zeta Function ◽  
Critical Line ◽  
Probabilistic Methods ◽  
Upper And Lower Bounds ◽  
Partial Sum ◽  
The Riemann Zeta Function ◽  
Order Of Magnitude ◽  
Riemann Zeta

Abstract The $2 q$-th pseudomoment $\Psi _{2q,\alpha }(x)$ of the $\alpha $-th power of the Riemann zeta function is defined to be the $2 q$-th moment of the partial sum up to $x$ of $\zeta ^\alpha $ on the critical line. Using probabilistic methods of Harper, we prove upper and lower bounds for these pseudomoments when $q \le \frac{1}{2}$ and $\alpha \ge 1$. Combined with results of Bondarenko et al., these bounds determine the size of all pseudomoments with $q> 0$ and $\alpha \ge 1$ up to powers of $\log \log x$, where $x$ is the length of the partial sum, and it turns out that there are three different ranges with different growth behaviours. In particular, the results give the order of magnitude of $\Psi _{2 q, 1}(x)$ for all $q> 0$.

A RELATION BETWEEN THE ZEROS OF TWO DIFFERENT L-FUNCTIONS WHICH HAVE AN EULER PRODUCT AND FUNCTIONAL EQUATION

2005 ◽  
Vol 01 (03) ◽  
pp. 401-429
Author(s):  
MASATOSHI SUZUKI
Keyword(s):  
Functional Equation ◽  
Zeta Function ◽  
Riemann Zeta Function ◽  
Modular Forms ◽  
Functional Equations ◽  
Euler Product ◽  
The Riemann Zeta Function ◽  
Euler Products ◽  
Riemann Zeta ◽  
Special Case

As automorphic L-functions or Artin L-functions, several classes of L-functions have Euler products and functional equations. In this paper we study the zeros of L-functions which have Euler products and functional equations. We show that there exists a relation between the zeros of the Riemann zeta-function and the zeros of such L-functions. As a special case of our results, we find relations between the zeros of the Riemann zeta-function and the zeros of automorphic L-functions attached to elliptic modular forms or the zeros of Rankin–Selberg L-functions attached to two elliptic modular forms.

Asymptotic estimates for Stieltjes constants: a probabilistic approach

2010 ◽  
Vol 467 (2128) ◽  
pp. 954-963 ◽  
Author(s):  
José A. Adell
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Differential Calculus ◽  
Probabilistic Approach ◽  
Upper Bounds ◽  
Asymptotic Estimates ◽  
Stieltjes Constants ◽  
The Riemann Zeta Function ◽  
Order Of Magnitude ◽  
Riemann Zeta

Let ( γ n ) n ≥ 0 be the sequence of Stieltjes constants appearing in the Laurent expansion of the Riemann zeta function. We obtain explicit upper bounds for | γ n |, whose order of magnitude is as n tends to infinity. To do this, we use a probabilistic approach based on a differential calculus for the gamma process.

scholarly journals Divergent Integrals, The Riemann Zeta Function, and The Vacuum

2020 ◽  
Author(s):  
Siamak Tafazoli
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Vacuum Energy ◽  
Opposite Sign ◽  
Vacuum Energy Density ◽  
Power Functions ◽  
Vacuum States ◽  
The Riemann Zeta Function ◽  
Order Of Magnitude ◽  
Riemann Zeta

This paper presents a new estimate for the vacuum energy density by summing the contributions of all quantum fields' vacuum states which turns out to be in the same order of magnitude (but with opposite sign) as the predictions of current cosmological models and all observational data to date. The basis for this estimate is the recent results on the analytical solution to improper integral of divergent power functions using the Riemann Zeta function.

scholarly journals Proof of the functional equation for the Riemann zeta-function

2022 ◽  
Vol Volume 44 - Special... ◽  
Author(s):  
Jay Mehta ◽  
P. -Y Zhu
Keyword(s):  
Functional Equation ◽  
Zeta Function ◽  
Riemann Zeta Function ◽  
Infinite Product ◽  
Partial Fraction ◽  
Partial Fraction Expansion ◽  
The Riemann Zeta Function ◽  
Euler Products ◽  
Riemann Zeta ◽  
Sine Functions

In this article, we shall prove a result which enables us to transfer from finite to infinite Euler products. As an example, we give two new proofs of the infinite product for the sine function depending on certain decompositions. We shall then prove some equivalent expressions for the functional equation, i.e. the partial fraction expansion and the integral expression involving the generating function for Bernoulli numbers. The equivalence of the infinite product for the sine functions and the partial fraction expansion for the hyperbolic cotangent function leads to a new proof of the functional equation for the Riemann zeta function.

Universality for generalized Euler products

Analysis ◽  
2006 ◽  
Vol 26 (3) ◽  
Author(s):  
Rasa Steuding
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Riemann Hypothesis ◽  
Equivalent Formulation ◽  
The Riemann Zeta Function ◽  
Euler Products ◽  
Riemann Zeta

In the paper, following ideas of Bohr and Bagchi, we present a new equivalent formulation of the Riemann hypothesis for the Riemann zeta-function ζ(

Universality of $L$-Dirichlet functions and nontrivial zeros of the Riemann zeta-function

2019 ◽  
Vol 210 (12) ◽  
pp. 1753-1773 ◽  
Author(s):  
A. Laurinčikas ◽  
J. Petuškinaitė
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Nontrivial Zeros ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

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 .

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