scholarly journals A Probabilistic Proof for Representations of the Riemann Zeta Function

Mathematics ◽  
2019 ◽  
Vol 7 (4) ◽  
pp. 369
Author(s):  
Jiamei Liu ◽  
Yuxia Huang ◽  
Chuancun Yin
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Probabilistic Approach ◽  
Recurrence Formula ◽  
Integral Representations ◽  
The Riemann Zeta Function ◽  
Probabilistic Proof ◽  
Riemann Zeta ◽  
Positive Integers

In this paper, we present a different proof of the well known recurrence formula for the Riemann zeta function at positive even integers, the integral representations of the Riemann zeta function at positive integers and at fractional points by means of a probabilistic approach.

scholarly journals LOG-TANGENT INTEGRALS AND THE RIEMANN ZETA FUNCTION

2019 ◽  
Vol 24 (3) ◽  
pp. 404-421
Author(s):  
Lahoucine Elaissaoui ◽  
Zine El-Abidine Guennoun
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Integrable Function ◽  
Harmonic Series ◽  
Algebraic Properties ◽  
Tangent Function ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Square Integrable Function ◽  
Positive Integers

We show that integrals involving the log-tangent function, with respect to any square-integrable function on (0,π/2), can be evaluated by the harmonic series. Consequently, several formulas and algebraic properties of the Riemann zeta function at odd positive integers are discussed. Furthermore, we show among other things, that the log-tangent integral with respect to the Hurwitz zeta function defines a meromorphic function and its values depend on the Dirichlet series ζh(s) :=∑n≥1hnn−s−8, where hn=∑nk=1(2k−1)−1.

scholarly journals IRRATIONALITÉ AUX ENTIERS IMPAIRS POSITIFS D'UN q-ANALOGUE DE LA FONCTION ZÊTA DE RIEMANN

2010 ◽  
Vol 06 (05) ◽  
pp. 959-988 ◽  
Author(s):  
FRÉDÉRIC JOUHET ◽  
ELIE MOSAKI
Keyword(s):  
Lower Bound ◽  
Vector Space ◽  
Zeta Function ◽  
Recent Result ◽  
Riemann Zeta Function ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Positive Integers

Dans cet article, nous nous intéressons à un q-analogue aux entiers positifs de la fonction zêta de Riemann, que l'on peut écrire pour s ∈ ℕ* sous la forme ζq(s) = ∑k≥1qk∑d|kds-1. Nous donnons une nouvelle minoration de la dimension de l'espace vectoriel sur ℚ engendré, pour 1/q ∈ ℤ\{-1; 1} et A entier pair, par 1, ζq(3), ζq(5), …, ζq(A - 1). Ceci améliore un résultat récent de Krattenthaler, Rivoal et Zudilin ([13]). En particulier notre résultat a pour conséquence le fait que pour 1/q ∈ ℤ\{-1; 1}, au moins l'un des nombres ζq(3), ζq(5), ζq(7), ζq(9) est irrationnel. In this paper, we focus on a q-analogue of the Riemann zeta function at positive integers, which can be written for s ∈ ℕ* by ζq(s) = ∑k≥1qk∑d|kds-1. We give a new lower bound for the dimension of the vector space over ℚ spanned, for 1/q ∈ ℤ\{-1; 1} and an even integer A, by 1, ζq(3), ζq(5), …, ζq(A-1). This improves a recent result of Krattenthaler, Rivoal and Zudilin ([13]). In particular, a consequence of our result is that for 1/q ∈ ℤ\{-1; 1}, at least one of the numbers ζq(3), ζq(5), ζq(7), ζq(9) is irrational.

scholarly journals Discrete universality of the Riemann zeta-function in short intervals

2020 ◽  
pp. 19-19
Author(s):  
Antanas Laurincikas
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Analytic Functions ◽  
Short Intervals ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Discrete Universality ◽  
Positive Integers

We consider the approximation of analytic functions by shifts of the Riemann zeta-function ?(s+ikh) with fixed h > 0 when positive integers k run over the interval [N,N+M], where N1/3(logN)26=15 ? M ? N, and prove that those k have a positive lower density as N ? ?. The same is true for some compositions. Two types of h > 0 are discussed separately.

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 Applications of uniform distribution theory to the Riemann zeta-function

2015 ◽  
Vol 64 (1) ◽  
pp. 67-74
Author(s):  
Selin Selen Özbek ◽  
Jörn Steuding
Keyword(s):  
Zeta Function ◽  
Uniform Distribution ◽  
Half Plane ◽  
Riemann Zeta Function ◽  
Distribution Theory ◽  
The Riemann Zeta Function ◽  
Upper Half Plane ◽  
Riemann Zeta ◽  
Positive Integers

Abstract We give two applications of uniform distribution theory to the Riemann zeta-function. We show that the values of the argument of are uniformly distributed modulo , where P(n) denotes the values of a polynomial with real coefficients evaluated at the positive integers. Moreover, we study the distribution of arg modulo π, where γn is the nth ordinate of a zeta zero in the upper half-plane (in ascending order).

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