Distribution modulo one and denominators of the Bernoulli polynomials

TL;DR

This paper establishes a precise link between the fractional parts of scaled integers and the prime divisors of the denominators of Bernoulli polynomials, revealing a new number-theoretic characterization.

Contribution

It provides a novel characterization connecting fractional parts of scaled integers to the prime divisors of Bernoulli polynomial denominators.

Findings

For any prime p, the sum of fractional parts exceeds 1 if and only if p divides the denominator of the Bernoulli polynomial.

The denominator of the Bernoulli polynomial without constant term is squarefree.

A one-to-one correspondence between fractional parts sum and prime divisibility is proven.

Abstract

Let $\{\cdot\}$ denote the fractional part and $n \geq 1$ be a fixed integer. In this short note, we show for any prime $p$ the one-to-one correspondence $$\sum_{\nu \geq 1} \left\{\frac{n}{p^\nu}\right\} > 1 \quad \iff \quad p \mid \mathrm{denom}( B_n(x) - B_n ),$$ 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.