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arXiv · 1708.07119

Distribution modulo one and denominators of the Bernoulli polynomials

Bernd C. Kellner
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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.

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Bernd C. Kellner. 2017-08-23. Distribution modulo one and denominators of the Bernoulli polynomials. https://arxiv.org/abs/1708.07119

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