Denominators of Bernoulli polynomials

Abstract

For a positive integer $n$ let ${\mathfrak{P}}_n={\prod }_{s_p(n)\ge p}p,$ where $p$ runs over all primes and $s_p(n)$ is the sum of the base $p$ digits of $n$. For all $n$ we prove that ${\mathfrak{P}}_n$ is divisible by all "small" primes with at most one exception. We also show that ${\mathfrak{P}}_n$ is large, has many prime factors exceeding $\sqrt{n}$, with the largest one exceeding $n^{20/37}$. We establish Kellner's conjecture, which says that the number of prime factors exceeding $\sqrt{n}$ grows asymptotically as $\kappa \sqrt{n}/\log n$ for some constant $\kappa$ with $\kappa =2$. Further, we compare the sizes of ${\mathfrak{P}}_n$ and ${\mathfrak{P}}_{n+1}$, leading to the somewhat surprising conclusion that although ${\mathfrak{P}}_n$ tends to infinity with $n$, the inequality ${\mathfrak{P}}_n>{\mathfrak{P}}_{n+1}$ is more frequent than its reverse.

25 pages

References in corpus (2)

• On a product of certain primes
• Power-Sum Denominators

Cited by in corpus (1)

• Distribution modulo one and denominators of the Bernoulli polynomials