Related papers: A third-order Apery-like recursion for $\zeta(5)$

We evaluate in closed form three special classes of alternating zeta series with one and two additional parameters. Two classes are expressed as linear combinations of polylogarithms while for the third class we prove an expression…

This manuscript presents a novel and reliable third-order iterative procedure for computing the zeros of solutions to second-order ordinary differential equations. By approximating the solution of the related Riccati differential equation…

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:…

We prove that among 1 and the odd zeta values $\zeta(3)$, $\zeta(5)$, \ldots, $\zeta(s)$, at least $ 0.21 \sqrt{s}/\sqrt{\log s}$ are linearly independent over the rationals, for any sufficiently large odd integer $s$. This is the first…

The Ap\'ery numbers may be defined by a cubic three-term recurrence relation, that is, a three-term relation where the coefficients are polynomials in the index of degree $3$. In this work, we first provide a systematic review of Ap\'ery…

Let $\zeta(s)$ be the Riemann zeta function. We prove the statement in the title, which improves a recent result of Rivoal and Zudilin by lowering $69$ to $35$. We also prove that at least one of $\beta(2),\beta(4),\ldots,\beta(10)$ is…

This work derives 5 methods to evaluate families of odd zeta values by combining a power of $\pi$ with Lambert series whose ratios of successive terms tend to $e^{-\pi\sqrt{a}}$ with integers $a\ge7$, outperforming Ramanujan's results with…

For a given irrational number, we consider the properties of best rational approximations of given parities. There are three different kinds of rational numbers according to the parity of the numerator and denominator, say odd/odd, even/odd…

We prove that the partial zeta function introduced in [9] is a rational function, generalizing Dwork's rationality theorem.

It is commonly known that $\zeta(2k) = q_{k}\frac{\zeta(2k + 2)}{\pi^2}$ with known rational numbers $q_{k}$. In this work we construct recurrence relations of the form $\sum_{k = 1}^{\infty}r_{k}\frac{\zeta(2k + 1)}{\pi^{2k}} = 0$ and show…

Using WZ pairs we present an infinite family of accelerated series for computing $\zeta(3)$.

In this note we show how the irrationality measure of $\zeta(s) = \pi^2/6$ can be used to obtain explicit lower bounds for $\pi(x)$. We analyze the key ingredients of the proof of the finiteness of the irrationality measure, and show how to…

Rational solutions of the fourth order analogue to the Painlev'e equations are classified. Special polynomials associated with the rational solutions are introduced. The structure of the polynomials is found. Formulas for their coefficients…

Many theorems of mathematics have the form that for a certain problem, e.g. a differential equation or polynomial (in)equality, there exists a solution. The sequential version then states that for a sequence of problems, there is a sequence…

In this paper, we establish improved effective irrationality measures for certain numbers of the form $\sqrt[3]{n}$, using approximations obtained from hypergeometric functions. These results are very close to the best possible using this…

It is well-known that separation of variables in 2nd order partial differential equations (PDEs) for physical problems with spherical symmetry usually leads to Cauchy's differential equation for the radial coordinate and Legendre's…

We present results for infinite series appearing in Feynman diagram calculations, many of which are similar to the Euler series. These include both one-, two- and three-dimensional series. All these series can be expressed in terms of…

The author proposed an elementary derivation of the formula of Yu.V. Nesterenko expansion in continued fraction of a number 2*$\zeta(3)$.

In 1997 the author found a criterion for the Riemann hypothesis for the Riemann zeta function, involving the nonnegativity of certain coefficients associated with the Riemann zeta function. In 1999 Bombieri and Lagarias obtained an…

We give a proof of the irrationality of the $p$-adic zeta-values $\zeta_p(k)$ for $p=2,3$ and $k=2,3$. Such results were recently obtained by F.Calegari as an application of overconvergent $p$-adic modular forms. In this paper we present an…