A note on numerators of Bernoulli numbers
arithmetic theory of algebraic function fields
In this note some observations on Bernoulli numbers are given in relation to function field analogs, some `known' counterexamples to a conjecture of Chowla have been pointed out. Consider \(z/(e^z-1)=\sum B_n z^n/n!\). The author handles the numerator \(N_n\) of \(B_n/n\). Ramanujan conjectured that \(N_n\) was always a prime. But 37 is a proper divisor of \(N_{32}\). There are other congruences and conjectures dealt with. The author mentions in particular the following (paraphrased) theorems.NEWLINENEWLINE NEWLINE1) The numerator \(N_n\) is not always squarefree; there are infinitely many examples of \(N_n\) being not squarefree.NEWLINENEWLINE NEWLINE2) For any given \(p\) of \(\mathbb F_q[t]\) and any positive integer \(k\), there are infinitely many \(n\) such that \(p^k\) divides the numerator \(N_n\) of \(B(n-1)!_c/n!_c\), where \(z/e(z)= \sum B_nz^n/n!_c\) with \(n!_c:= \prod D_i^{n_i}\in \mathbb F_q[t]\), where \(D_m:= \prod^{m-1}_{i=0} (t^{q^m}- t^{q^i})\).NEWLINENEWLINE NEWLINE3) Without going into details (v.d.W.), the author gives the result: Given any ``irregular'' prime \(p\) defined as in the last paragraph, and any positive integer \(k\), there are infinitely many \(n\) such that ``\(\beta(n)\)'' is divisible by \(p^k\) (view the detail in the paper).NEWLINENEWLINE NEWLINEAs such the paper offers an outlook on conjectures of others, like work of Kummer, Herbrandt-Ribet, Goss, Bernoulli-Carlitz numbers, Siegel, Chowla, von Staudt, Buhler and Harvey, Ernvall and Metsänkylä, et al. The author also gives connections and applications of Bernoulli numbers; see the first paragraph of the paper.
scientific article; zbMATH DE number 6204239