A variation of Euler's approach to values of the Riemann zeta function
  An elementary method of computing the values at negative integers of the Riemann zeta function is presented. The principal ingredient is a new q-analogue of the Riemann zeta function. We show that for any argument other than 1 the classical limit of this q-analogue exists and equals the value of the Riemann zeta. 
Masanobu Kaneko, Nobushige Kurokawa and Masato Wakayama