A $q$-analogue for Euler's evaluations of the Riemann zeta function

Abstract

We provide a $q$-analogue of Euler's formula for $\zeta(2k)$ for $k\in\mathbb{Z}^+$. Our main results are stated in Theorems 3.1 and 3.2 below. The result generalizes a recent result of Z.W. Sun who obtained $q$-analogues of $\zeta(2)=\pi^2/6$ and $\zeta(4)=\pi^4/90$.