The universal von Staudt theorems

Abstract

Abstract.We prove general forms of von Staudt's theorems on the Bernoulli numbers. As a consequence we are able to deduce strong versions of a number of congruences involving various generalisations of the Bernoulli numbers. For example we obtain an improved form of a congruence due to Hurwitz involving the Laurent series coefficients of the Weierstrass elliptic function associated with a square lattice.

Hyperfunctions, formal groups and generalized Lipschitz summation formulas

“…The corresponding numbers by construction coincide with the universal Bernoulli numbers discovered in [5], and generated by…”

“…Many generalizations of this result have been obtained in the literature in the last decades. In an attempt to clarify the deep connection between these congruences and algebraic topology, in [5] Clarke proposed the notion of universal Bernoulli numbers B n , defined as (50), and proved the remarkable universal von Staudt congruence. If n is even, we have Like the classical ones, the universal Bernoulli numbers as well play an important role in several branches of mathematics, in particular in complex cobordism theory (see e.g.…”

The Lazard formal group, universal congruences and special values of zeta functions

“…The universal Bernoulli numbers satisfy some very interesting congruences. In particular, we will discuss briefly the universal Von Staudt's congruence [13] (that generalizes the classical Clausen-von Staudt congruence and several variations of it known in the literature) and the universal Kummer congruence [1]. Both congruences have a distinguished role in algebraic geometry [7], [20].…”

“…The congruences (13) and (14) hold due to the universal congruences (6) and (7), and by taking into account that B G 0 = 1. Let us assume that c p−1 ≡ 1 mod p for p odd (the case c p−1 ≡ 0 mod p will follow immediately).…”

“…Concerning the congruence (13), in the r.h.s. we can suppose that in each addend of the first sum p ∤ k; otherwise the addend would contribute an integer.…”