Treffer: On mth order Bernoulli polynomials of degree m that are Eisenstein: On \(m\)th order Bernoulli polynomials of degree~\(m\) that are Eisenstein

The Bernoulli polynomials \(B_m^l(x)\in{\mathbb Q}[x]\) \((m\geq 0)\) of order \(l\) are defined by the generating function \((t/(e^t-1))^le^{tx}\). The authors are concerned with these polynomials for \(l=m\) as \(m\to\infty\). They show that asymptotically more than \(1/5\) of the polynomials \(B_m^m(x)\) satisfy the Eisenstein irreducibility criterion for some prime \(p\) (depending on \(m\)). It was shown by the first author [J. Number Theory 59, 374-388 (1997; Zbl 0866.11013)] that this problem can be reduced to a divisibility property of ordinary Bernoulli numbers. The authors study this property by using known facts about Bernoulli numbers, including a relationship of these numbers to the Riemann zeta function.