Related papers: Irrationality of some p-adic L-values
An important component of Ap\'ery's proof that $\zeta (3)$ is irrational involves representing $\zeta (3)$ as the limit of the quotient of two rational solutions to a three-term recurrence. We present various approaches to such Ap\'ery…
In this article, we present a new linear independence criterion for values of the $p$-adic polygamma functions defined by J.~Diamond. As an application, we obtain the linear independence of some families of values of the $p$-adic Hurwitz…
We show how one can use Hermite-Pad\'{e} approximation and little $q$-Jacobi polynomials to construct rational approximants for $\zeta_q(2)$. These numbers are $q$-analogues of the well known $\zeta(2)$. Here $q=\frac{1}{p}$, with $p$ an…
In [Pollack-Stevens 2011], efficient algorithms are given to compute with overconvergent modular symbols. These algorithms then allow for the fast computation of $p$-adic $L$-functions and have further been applied to compute rational…
Building upon ideas of the second and third authors, we prove that at least $2^{(1-\varepsilon)\frac{\log s}{\log\log s}}$ values of the Riemann zeta function at odd integers between 3 and $s$ are irrational, where $\varepsilon$ is any…
In this paper, we will show that the $p$-adic valuation (where $p$ is a given prime number) of some type of rational numbers is unusually large. This generalizes the very recent results by the author and by A. Dubickas, which are both…
This note proves that the first odd zeta value does not have a closed form formula $\zeta(3)\ne r \pi^3$ for any rational number $r \in \mathbb{Q}$. Furthermore, assuming the irrationality of the second odd zeta value $\zeta(5)$, it is…
We compute the $p$-adic $L$-functions of evil Eisenstein series, showing that they factor as products of two Kubota--Leopoldt $p$-adic $L$-functions times a logarithmic term. This proves in particular a conjecture of Glenn Stevens.
We compute the Archimedean doubling zeta integrals which appear in the interpolation formulas for the p-adic L-functions of Siegel modular forms, and verify that they agree with the modified Archimedean Euler factors for p-adic…
In this note we evaluate multiple integrals that play a crucial role in the theory of irrationality of zeta function
In a recent paper with Sprang and Zudilin, the following result was proved: if $a$ is large enough in terms of $\varepsilon>0$, then at least $2^{(1-\varepsilon)\frac{\log a}{\log \log a}}$ values of the Riemann zeta function at odd…
A failed attempt to prove the universality of Lerch zeta function $L(\lambda,\alpha,s)$ when $\lambda$ is irrational and $\alpha$ is rational, and for any $\lambda$ when $\alpha$ is irrational algebraic.
It is proved that, for all odd integer $s \geqslant s_0(\varepsilon)$, there are at least $\big( c_0 - \varepsilon \big) \frac{s^{1/2}}{(\log s)^{1/2}} $ many irrational numbers among the following odd zeta values:…
A general hypergeometric construction of linear forms in (odd) zeta values is presented. The construction allows to recover the records of Rhin and Viola for the irrationality measures of $\zeta(2)$ and $\zeta(3)$, as well as to explain…
We explain in detail how to accelerate continued fractions (for constants as well as for functions) using the method used by R.~Ap\'ery in his proof of the irrationality of $\zeta(3)$. We show in particular that this can be applied to a…
In this note, we prove the irrationality of $\zeta(5)$ and generalize the method to prove the irrationality of all higher odd zeta values. Our proof relies on the method of contradiction, existence of solution of a system of Linear…
New (infinitely many) rational approximants to \zeta(3) proving its irrationality are given. The recurrence relations for the numerator and denominator of these approximants as well as their continued fraction expansions are obtained. A…
We exploit some properties of the Hurwitz zeta function $\zeta (n,x)$ in order to study sums of the form $\frac{1}{\pi ^{n}}\sum_{j=-\infty}^{\infty}1/(jk+l)^{n}$ and $\frac{1}{\pi ^{n}}\sum_{j=-\infty}^{\infty}(-1)^{j}/(jk+l)^{n}$ for $%…
We study zeta functions enumerating finite-dimensional irreducible complex linear representations of compact p-adic analytic and of arithmetic groups. Using methods from p-adic integration, we show that the zeta functions associated to…
The Euler--Riemann zeta function is a largely studied numbertheoretic object, and the birthplace of several conjectures, such as the Riemann Hypothesis. Different approaches are used to study it, including $p$-adic analysis : deriving…