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 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 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.

scholarly journals Some local-convexity theorems for the zeta-function-like analytic functions

1988 ◽  
Vol Volume 11 ◽  
Author(s):  
R Balasubramanian ◽  
K Ramachandra
Keyword(s):  
Lower Bound ◽  
Zeta Function ◽  
Lower Bounds ◽  
Exponential Function ◽  
Riemann Zeta Function ◽  
Analytic Functions ◽  
Local Convexity ◽  
The Riemann Zeta Function ◽  
International Audience ◽  
Riemann Zeta

International audience In this paper we investigate lower bounds for $$I(\sigma)= \int^H_{-H}\vert f(\sigma+it_0+iv)\vert^kdv,$$ where $f(s)$ is analytic for $s=\sigma+it$ in $\mathcal{R}=\{a\leq\sigma\leq b, t_0-H\leq t\leq t_0+H\}$ with $\vert f(s)\vert\leq M$ for $s\in\mathcal{R}$. Our method rests on a convexity technique, involving averaging with the exponential function. We prove a general lower bound result for $I(\sigma)$ and give an application concerning the Riemann zeta-function $\zeta(s)$. We also use our methods to prove that large values of $\vert\zeta(s)\vert$ are ``rare'' in a certain sense.

scholarly journals More than five-twelfths of the zeros of $$\zeta $$ are on the critical line

2019 ◽  
Vol 7 (1) ◽  
Author(s):  
Kyle Pratt ◽  
Nicolas Robles ◽  
Alexandru Zaharescu ◽  
Dirk Zeindler
Keyword(s):  
Number Theory ◽  
Lower Bound ◽  
Zeta Function ◽  
Riemann Zeta Function ◽  
Critical Line ◽  
Dirichlet Polynomial ◽  
Real Numbers ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Combinatorial Process

AbstractThe second moment of the Riemann zeta-function twisted by a normalized Dirichlet polynomial with coefficients of the form $$(\mu \star \Lambda _1^{\star k_1} \star \Lambda _2^{\star k_2} \star \cdots \star \Lambda _d^{\star k_d})$$(μ⋆Λ1⋆k1⋆Λ2⋆k2⋆⋯⋆Λd⋆kd) is computed unconditionally by means of the autocorrelation of ratios of $$\zeta $$ζ techniques from Conrey et al. (Proc Lond Math Soc (3) 91:33–104, 2005), Conrey et al. (Commun Number Theory Phys 2:593–636, 2008) as well as Conrey and Snaith (Proc Lond Math Soc 3(94):594–646, 2007). This in turn allows us to describe the combinatorial process behind the mollification of $$\begin{aligned} \zeta (s) + \lambda _1 \frac{\zeta '(s)}{\log T} + \lambda _2 \frac{\zeta ''(s)}{\log ^2 T} + \cdots + \lambda _d \frac{\zeta ^{(d)}(s)}{\log ^d T}, \end{aligned}$$ζ(s)+λ1ζ′(s)logT+λ2ζ′′(s)log2T+⋯+λdζ(d)(s)logdT,where $$\zeta ^{(k)}$$ζ(k) stands for the kth derivative of the Riemann zeta-function and $$\{\lambda _k\}_{k=1}^d$${λk}k=1d are real numbers. Improving on recent results on long mollifiers and sums of Kloosterman sums due to Pratt and Robles (Res Number Theory 4:9, 2018), as an application, we increase the current lower bound of critical zeros of the Riemann zeta-function to slightly over five-twelfths.

scholarly journals Many odd zeta values are irrational

2019 ◽  
Vol 155 (5) ◽  
pp. 938-952 ◽  
Author(s):  
Stéphane Fischler ◽  
Johannes Sprang ◽  
Wadim Zudilin
Keyword(s):  
Lower Bound ◽  
Zeta Function ◽  
Real Number ◽  
Riemann Zeta Function ◽  
Positive Real Number ◽  
Positive Real ◽  
Linear Forms ◽  
The Riemann Zeta Function ◽  
Zeta Values ◽  
Riemann Zeta

Building upon ideas of the second and third authors, we prove that at least$2^{(1-\unicode[STIX]{x1D700})(\log s)/(\text{log}\log s)}$values of the Riemann zeta function at odd integers between 3 and$s$are irrational, where$\unicode[STIX]{x1D700}$is any positive real number and$s$is large enough in terms of$\unicode[STIX]{x1D700}$. This lower bound is asymptotically larger than any power of$\log s$; it improves on the bound$(1-\unicode[STIX]{x1D700})(\log s)/(1+\log 2)$that follows from the Ball–Rivoal theorem. The proof is based on construction of several linear forms in odd zeta values with related coefficients.

Sign in / Sign up

Export Citation Format

Share Document