Related papers: A closed-form expression for $\zeta(3)$

In this note, using an idea from \cite{Amo-Carrillo-Sanchez} we derive some new series representations involving $\zeta(2n)$ and Euler numbers. Using a well-known series representation for the Clausen function, we also provide some new…

Already in 1734 Euler found a short explicit formula for the value of Riemann zeta function Zeta(s) when the argument s equals a positive integer 2n where n=1,2,3,. No such formula exists for odd positive integer arguments of Zeta. The…

We give analogs for Euler's constant and ln(4/Pi) of the well-known double integrals for zeta(2) and zeta(3). We also give a series for ln(4/Pi) which reveals it to be an "alternating Euler constant."

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…

New expansions of the number zeta(3) in continuous fractions are found.

Based on the framework of the WZ theory, a new evaluation for $\varsigma (2) = \frac{\pi ^2}{6}$ and $\varsigma (4) = \frac{\pi ^4}{90}$ was given respectively, finally, a new recurrence formula for $\varsigma (2k)$ was given.

Using a summation identity obtained for the Fourier coefficients of $x^{2k}$, we derive a closed form expression for the zeta function at even positive integers, using a technique similar to one in an existing proof by Aladdi and Defant[1],…

We obtain the last of the standard Kuznetsov formulas for $SL(3,\Bbb{Z})$. In the previous paper, we were able to exploit the relationship between the positive-sign Bessel function and the Whittaker function to apply Wallach's Whittaker…

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

Beuker's [2] considers the following integral $$ \int_{0}^{1}\int_{0}^{1} \frac{-\log xy}{1-xy} P_n(x)P_n(y)\ dx dy$$If $d_n=\text{LCM}(1,2,...,n)$, then $$ 0<\frac{|A_n+B_n\zeta(3)|}{d_n^3}<2(\sqrt{2}-1)^{4n} \zeta(3) $$ for some…

Let $\mathcal{B}$ be a compact convex planar domain with smooth boundary of finite type and $\mathcal{B}_\theta$ its rotation by an angle $\theta$. We prove that for almost every $\theta\in[0, 2\pi]$ the remainder…

The relative density of visible points of the integer lattice $\mathbb{Z}^d$ is known to be $1/\zeta(d)$ for $d\geq 2$, where $\zeta$ is Riemann's zeta function. In this paper we prove that the relative density of visible points in the…

We present new sharper lower and upper bounds for the non-zero Bernoulli numbers using Euler's formula for the Riemann zeta function. In particular, we determine the best possible constants $ \alpha $ and $ \beta $ such that the double…

We give an explicit formula for the subalgebra zeta function of a general 3-dimensional Lie algebra over the p-adic integers $\mathbb{Z}_p$. To this end, we associate to such a Lie algebra a ternary quadratic form over $\mathbb{Z}_p$. The…

In this sequel to our recent note it is shown, in a unified manner, by making use of some basic properties of certain special functions, such as the Hurwitz zeta function, Lerch zeta function and Legendre chi function, that the values of…

In a recent work, Dancs and He found an Euler-type formula for $\,\zeta{(2\,n+1)}$, $\,n\,$ being a positive integer, which contains a series they could not reduce to a finite closed-form. This open problem reveals a greater complexity in…

We provide rapidly converging formulae for the Riemann zeta function at odd integers using the Lambert series $\mathscr{L}_q(s) = \sum_{n=1}^\infty n^{s} q^{n}/(1-q^n)$, $s=-(4k\pm 1)$. Our main formula for $\zeta(4k-1)$ converges at rate…

This study deals with certain harmonic zeta functions, one of them occurs in the study of the multiplication property of the harmonic Hurwitz zeta function. The values at the negative even integers are found and Laurent expansions at poles…

We determine the classical approximation constants $w_{3}(\zeta),w_{3}^{\ast}(\zeta),\lambda_{3}(\zeta)$ such as the uniform constants $\widehat{w}_{3}(\zeta),\widehat{w}_{3}^{\ast}(\zeta),\widehat{\lambda}_{3}(\zeta)$ associated to real…

Let $\pi$ be a Hecke-Maass cusp form for $SL(3,\mathbb Z)$. In this paper we will prove the following subconvex bound $$ L(\tfrac{1}{2}+it,\pi)\ll_{\pi,\varepsilon} (1+|t|)^{3/4-1/16+\varepsilon}. $$