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#### Congruence among generalized Bernoulli numbers.

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##### Citation

Szmidt, J., Urbanowicz, J., & Zagier, D. (1995). Congruence among generalized Bernoulli
numbers.* Acta Arithmetica,* *71*(3), 273-278.

Cite as: https://hdl.handle.net/21.11116/0000-0004-38F5-9

##### Abstract

Let χ denote a primitive quadratic character mod M (or the trivial character) and let d be a fundamental discriminant (or 1). Denote by χ' the character mod M |d | induced by χ. The authors consider the generalized Bernoulli numbers Bm, χ' and the corresponding Bernoulli polynomials Bm, χ' (X) at X = d. From a certain set of these numbers they form an expression P(d,m), too complicated to reproduce here, which is in fact a linear combination (with integer coefficients) of Bm, χ' and Bm, χ' (d), where m is a fixed integer ≥ 1. The expression also depends upon two integral parameters to be chosen under some restrictions. The authors prove that P(d,m) \in 2^h 3^km \bbfZ, where h and k are given nonnegative integers. As a consequence one has a number of divisibility statements for Bm, χ/m and so for the class numbers of quadratic fields. Results of the same type, including special cases of the present result, have previously appeared in several papers [e.g. \it K. Hardy and \it K. S. Williams, Acta Arith. 47, 263-276 (1986; Zbl 0598.12003)].\par The proof involves some clever manipulations related to the series \sum^∞n = 1 χ (n) e^nt.