A Series of New Congruences for Bernoulli Numbers and Eisenstein Series Original Research Article

We prove congruences of shape Ek+h≡Ek·Eh (mod N) modulo powers N of small prime numbers p, thereby refining the well-known Kummer-type congruences modulo these p of the normalized Eisenstein series Ek. The method uses Serreʹs theory of Iwasawa functions and p-adic Eisenstein series; it presents a rather general procedure to find and verify such congruences with a modest amount of numerical calculation.