Distribution modulo one and denominators of the Bernoulli polynomials

Abstract

Let $\{\cdot \}$ denote the fractional part and $n\ge 1$ be a fixed integer. In this short note, we show for any prime $p$ the one-to-one correspondence $$\sum _{\nu \ge 1}\left\{\frac{n}{p^{\nu }}\right\}>1⟺p\mid \mathrm{denom}(B_n(x)-B_n),$$ where $B_n(x)-B_n$ is the $n$th Bernoulli polynomial without constant term and $\mathrm{denom}(\cdot )$ is its denominator, which is squarefree.

9 pages

References in corpus (1)

• On a product of certain primes