Bernoulliho čísla a regulární prvočísla

Abstract

The aim of this work is to study the relation between regular primes and regular Bernoulli numbers (or just simply Bernoulli numbers). By the class number formula we connect the class number to the values of Dirichlet L-series. We then compute certain values of Dirichlet L-series in terms of generalized Bernoulli numbers. In order to investigate the relations between two types of Bernoulli numbers we define the p-adic Dirichlet L-series. In the end we get a congruence between the class number and Bernoulli numbers modulo p. Since the regular primes are those which divide the corresponding class numbers this is precisely our goal. 1

Cı'lem pra'ce je studium vztahu mezi regula'nı'mi prvocˇı'sly a regula'rnı'mi Bernoulliho cˇı'sly (nebo jednodusě jen Bernoulliho cˇı'sly). Formulı' trˇı'dove'ho cˇı'sla spojı'me trˇı'dove' cˇı'slo s hodnotami Di- richletovy'ch L-rˇad. Pote' vypocťeme urcˇite' hodnoty Dirichletovy'ch L-rˇad pomocı' zobecneňy'ch Bernoulliho cˇı'sel. Abychom vysětrˇili vztahy mezi dveˇma typy Bernoulliho cˇı'sel, definujeme p- adicke' Dirichletovy L-rˇady. Na konci pra'ce dostaneme kongruenci mezi trˇı'dovy'm cˇı'slem a Ber- noulliho cˇı'sly modulo p. Z definice jsou regula'rnı' cˇı'sla pra'veˇ ta, ktera' deľı' prˇı'slusňa' trˇı'dova' cˇı'sla, a proto jsme dosa'hli sve'ho cı'le. 1