# The Hurwitz Zeta Function at the Positive Integers

## TL;DR

This paper derives simplified formulas for the Hurwitz zeta function at positive integers, providing real and imaginary parts separately and establishing an analytic continuation for its generating function.

## Contribution

It offers new, simpler formulas for the Hurwitz zeta function at positive integers and develops an analytic continuation for its generating function.

## Key findings

- Explicit formulas for real and imaginary parts of (k,b) at positive integers
- Analytic continuation of the generating function for (k,b)
- Simpler computational methods for (k,b) at positive integers

## Abstract

We address the problem of finding out the values of the Hurwitz zeta function at the positive integers $k$, $\zeta(k,b)$, by working out their real and imaginary parts separately and then combining them. A few different formulae for the Hurwitz zeta function are known from the literature, but they are very general and usually hold for $\Re{(k)}>1$. The advantage of formulae that only hold at the positive integers is the fact that they are usually simpler and easier to work with. We also obtain an analytic continuation for the generating function of $\zeta(k,b)$ as $\sum_{k\ge 2}x^k(\zeta(k,b)-1/b^k)$, valid for complex $x$ and $b$, where term $1/b^k$ was subtracted for convenience.