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 $\zeta (4)={\pi }^4/90$.

Submitted to Research in Number Theory