Related papers: Ramanujan's Formula for $\zeta(2n+1)$

For k <= n, let E(2n,k) be the sum of all multiple zeta values with even arguments whose weight is 2n and whose depth is k. Of course E(2n,1) is the value of the Riemann zeta function at 2n, and it is well known that E(2n,2) = (3/4)E(2n,1).…

An alternative formula is presented for the evaluation of the zeta function values $\zeta(2k)$ without the need for Bernoulli numbers. Our formula is recursive, and improves the efficiency with which we can calculate large values of the…

Ramanujan, in his famous first letter to Hardy, claimed a very precise estimate for the number of integers that can be written as a sum of two squares. Far less well-known is that he also made further claims of a similar nature for the…

We give some restricted sum formulas for double zeta values whose arguments satisfy certain congruence conditions modulo 2 or 6, and also give an application to identities showed by Ramanujan for sums of products of Bernoulli numbers with a…

At scattered places of his notebooks, Ramanujan recorded over 30 values of singular moduli $\alpha_n$. All those results were proved by Berndt et. al by employing Weber-Ramanujan's class invariants. In this paper, we initiate to derive the…

This paper is a tribute to the genius of the legendary Indian mathematician Srinivasa Ramanujan (22 December 1887 - 26 April 1920) in the centenary year of his death. The life story of Ramanujan is so well known that it needs no elaboration…

This paper presents a new approach to evaluating the special values of the Dirichlet beta function, $\beta(2k+1)$, where $k$ is any nonnegative integer. Our approach relies on some properties of the Euler numbers and polynomials, and uses…

Towards the end of his life Ramanujan wrote a manuscript on properties of the partition and tau functions, some parts of which remained unpublished until very recently. Nevertheless, this manuscript gave rise to a lot of subsequent work. In…

The cyclic insertion conjecture of Borwein, Bradley, Broadhurst and Lison\v{e}k states that inserting all cyclic shifts of some fixed blocks of 2's into the multiple zeta value {\zeta}(1,3,...,1,3) gives an explicit rational multiple of a…

Plouffe conjectured rapidly converging series formulas for $\pi^{2n+1}$ and $\zeta (2n+1)$ for small values of $n$. We find the general pattern for all nonnegative integer values of $n$ and offer a proof.

In 2001, Kanemitsu, Tanigawa, and Yoshimoto studied the following generalized Lambert series, $$ \sum_{n=1}^{\infty} \frac{n^{N-2h} }{\exp(n^N x)-1}, $$ for $N \in \mathbb{N}$ and $h\in \mathbb{Z}$ with some restriction on $h$. Recently,…

In $1735$ Euler \cite{1} proved that for each positive integer $k$, the series $\zeta(2k) = \sum_{\ell=1}^{\infty} \ell^{-2k}$ converges to a rational multiple of $\pi^{2k}$. Many demonstrations of this fact are now known, and Euler's…

This paper treats about one of the most remarkable achievements by Riemann, that is the symmetric form of the functional equation for {\zeta}(s). We present here, after showing the first proof of Riemann, a new, simple and direct proof of…

In this paper, we find rational zeta series with $\zeta(2n)$ in terms of $\zeta(2k+1)$ and $\beta(2k)$, the Dirichlet beta function. We then develop a certain family of generalized rational zeta series using the generalized Clausen function…

In the first part we establish a connection between the Euler-Maclaurin summation formula and the Rota-Baxter functional equation. In the second part we give a simple proof of a formula, due to Ramanujan, on the summation of certain…

In this paper we present a simple method for deriving an alternative form of the functional equation for Riemann's Zeta function. The connections between some functional equations obtained implicitly by Leonhard Euler in his work "Remarques…

In this short research note, we aim to establish an interesting extension of a summation due to Ramanujan.The result is derived with the help of an extension of Gauss's summation theorem available in the literature.

Recently, Simon Plouffe has discovered a number of identities for the Riemann zeta function at odd integer values. These identities are obtained numerically and are inspired by a prototypical series for Apery's constant given by Ramanujan:…

Let $\Bbb Z$ and $\Bbb Z^+$ be the set of integers and the set of positive integers, respectively. For $a,b,c,d,n\in\Bbb Z^+$ let $t(a,b,c,d;n)$ be the number of representations of $n$ by $ax(x+1)/2+by(y+1)/2+cz(z+1)/2+dw(w+1)/2$…

The values of the Riemann zeta function at odd positive integers, $\zeta(2n+1)$, are shown to admit a representation proportional to the finite-part of the divergent integral $\int_0^{\infty} t^{-2n-1} \operatorname{csch}t\,\mathrm{d}t$.…