The Hurwitz zeta function at the positive integers

Question

Is there a formula that gives the values ​​$\zeta(2n,a)$ as a function of $a$ and Bernoulli numbers, where $n$ is a natural number and $0<a≤1$?

$\zeta(z,a)$ is the Hurwitz zeta function.

Answer 1

You're looking at the wrong thing. The values of the zeta-function at negative integers are what are really related to Bernoulli numbers: $\zeta(1-n) = -B_n/n$ for $n \geq 2$. And this generalizes very cleanly to the Hurwitz zeta-function: for $n \geq 2$, $\zeta(1-n,a) = -{\bf B}_n(a)/n$ where ${\bf B}_n(x)$ is the $n$th Bernoulli polynomial.