Generalized Bernoulli-Hurwitz Numbers and The Universal Bernoulli Numbers

The three fundamental properties of the Bernoulli numbers, namely, the theorem of von Staudt-Clausen, von Staudt’s second theorem, and Kummer’s original congruence, are generalized to new numbers that we call generalized Bernoulli-Hurwitz numbers. These are coefficients of power series expansion of a higher genus algebraic function with respect to suitable variable. Our generalization strongly contrasts with the previous works. Indeed, the order of the power of the modulus prime in our Kummer-type congruences is exactly the same as in trigonometric function case, namely, Kummer’s own congruence for the original Bernoulli numbers, and as in elliptic function case, namely, H. Lang’s extension to the Hurwitz numbers. However, in the other past results on higher genus algebraic functions, the modulus was at most half of these classical cases. This contrast is clarified by investigating the analog of the three properties above for the universal Bernoulli numbers. (Accepted and Corrected Version : June 5, 2012)