p-adic zeta functions and Bernoulli numbers

One gives a new proof to the Leopoldt-Kubota-Iwasawa theorem regarding the possibility of the p-adic interpolation of the values of the Riemann zeta-function and of the Dirichlet L-functions at negative integral points. To this end, for each root ? ≠ 1 of unity one introduces and one investigates the numbers C_n(?) which arise in the expansion $$frac{{varepsilon - 1}}{{varepsilon e^z - 1}} = sumlimits_{n = 0}^infty {frac{{C_n (varepsilon )}}{{n!}}Z^n }$$ One proves a generalization of the Kummer congruences for the Bernoulli numbers.