A new formula for $\zeta(s)$

Abstract

In this paper, by introducing a new operation in the vector space of analytic functions, the author presents a method for derivating the well-known formulas: $\zeta(1-k)=-\frac{B_k}{k}$ and $\zeta(1-n,a)=-\frac{B_n(a)}{n}$ , where $\zeta$, $\zeta(1-n,a)$ denote the Riemann zeta function and the Hurwitz zeta function respectively. $B_k$ is the $k$-th Bernoulli number. Also the author steps further to deduce some identities related to Bernoulli number and Bernoulli polynomial. Moreover, when combining the operation with forward difference, we can show a new formula for Riemann zeta function, i.e. \[\zeta(s)=e\sum_{n=0}^{\infty}\sum_{i=0}^{n}(-1)^{n-i}\frac{1}{(n-i)!(1+i)^{s}}.\]