Generalized Bernoulli–Hurwitz numbers and the universal Bernoulli numbers

The three fundamental properties of the Bernoulli numbers, namely, the von Staudt–Clausen theorem, 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 in the power series expansion of a higher-genus algebraic function with respect to a suitable variable. Our generalization differs strongly from previous works. Indeed, the order of the power of the modulus prime in our Kummer-type congruences is exactly the same as in the trigonometric function case (namely, Kummer's own congruence for the original Bernoulli numbers), and as in the elliptic function case (namely, H. Lang's extension for the Hurwitz numbers). However, in other past results on higher-genus algebraic functions, the modulus was at most half of its value in these classical cases. This contrast is clarified by investigating the analogue of the three properties above for the universal Bernoulli numbers.
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