Journal of the
Korean Mathematical Society JKMS

In the complex case, we construct a $q$-analogue of the Riemann zeta function $\zeta_q(s)$ and a $q$-analogue of the Dirichlet $L$-function $L_q(s,\chi),$ which interpolate the $q$-analogue Bernoulli numbers. Using the properties of $p$-adic integrals and measures, we show that Kummer type congruences for the $q$-analogue Bernoulli numbers are the generalizations of the usual Kummer congruences for the ordinary Bernoulli numbers. We also construct a $q$-analogue of the $p$-adic $L$-function $L_{p}(s,\chi;q)$ which interpolates the $q$-analogue Bernoulli numbers at non positive integers.

Keywords: Bernoulli number, Kummer Congruence, $p$-adic $L$-function

MSC numbers: 11E95, 11M38