Related papers: On quotients of Riemann zeta values at odd and eve…

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 $\int_0^{\infty} t^{-2n-1} \operatorname{csch}t\,\mathrm{d}t$.…

Using Parseval's identity for the Fourier coefficients of $x^k$, we provide a new proof that $\zeta(2k)=\dfrac{(-1)^{k+1}B_{2k}(2\pi)^{2k}}{2(2k)!}$.

We investigate the values of the Riemann zeta function at odd integers and the Dirichlet beta function at even integers, by collecting several distinct analytic frameworks converging to these values, thus providing a unifying perspective.…

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 introduce and study "elliptic zeta values", a two-parameter deformation of the values of Riemann's zeta function at positive integers. They are essentially Taylor coefficients of the logarithm of the elliptic gamma function, and share…

Euler discovered a formula for expressing the value of the Riemann zeta function for all even positive integer arguments. A closed-form expression for the Riemann zeta function for all odd integer arguments, based on the values of the…

By using the generalized Bernoulli numbers, we deduce new integral representations for the Riemann zeta function at positive odd-integer arguments. The explicit expressions enable us to obtain criteria for the dimension of the vector space…

In this paper, we study a family of single variable integral representations for some products of $\zeta(2n+1)$, where $\zeta(z)$ is Riemann zeta function and $n$ is positive integer. Such representation involves the integral…

The functional relation of the Riemann z\^eta function provides us with neither the nature nor the expression of z\^eta at positive odd numbers. From the function $F(z)=\frac{z^{-2n}}{e^z-1}$, we find a functional relation involving…

We study the derivatives of polynomials with equally spaced zeros and find connections to the values of the Riemann zeta-function at the positive even integers.

Using a polylogarithmic identity, we express the values of $\zeta$ at odd integers $2n+1$ as integrals over unit $n-$dimensional hypercubes of simple functions involving products of logarithms. We also prove a useful property of those…

We propose a relation between values of the Riemann zeta function $\zeta$ and a family of integrals. This results in an integral representation for $\zeta(2p)$, where $p$ is a positive integer, and an expression of $\zeta(2p+1)$ involving…

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…

In this article, we derive a Euler prime product formula for the magnitude of the Riemann zeta function $\zeta(s)$ valid for $\Re(s)>1$, as well as similar formulas for $\zeta(s)$ valid for an even and odd $k$th positive integer argument.…

The author derives new family of series representations for the values of the Riemann Zeta function $\zeta(s)$ at positive odd integers. For $n\in\mathbb{N}$, each of these series representing $\zeta(2n+1)$ converges remarkably rapidly with…

In this paper, by introducing a new operation in the vector space of Laurent series, the author derived explicit series for the values of $\zeta$-funtion at positive integers, where $\zeta$ denotes the Riemann zeta function. The values of…

In this article, we explore a natural extension of the quadratic parametrization introduced in our previous work. By replacing the integer $n$ by $n^s$ ($ s\in\mathbb{R}, s>1$) and allowing the parameters to be real, we obtain for each…

We give a probabilistic interpretation of the Dedekind zeta functions of $\mathbb{Q}(\sqrt{-1})$ and $\mathbb{Q}(\sqrt{-2})$ using zeta integrals and use this to show that the first two Li coefficients of these zeta functions are positive.…

We show that the $n$th derivative of the Riemann zeta function, when summed over the non-trivial zeros of zeta, is real and positive/negative in the mean for $n$ odd/even, respectively. We show this by giving a full asymptotic expansion of…

A Ramanujan-type formula involving the squares of odd zeta values is obtained. The crucial part in obtaining such a result is to conceive the correct analogue of the Eisenstein series involved in Ramanujan's formula for $\zeta(2m+1)$. The…