Distribution modulo one and denominators of the Bernoulli polynomials

2017/08/23 by Bernd C. Kellner, Kellner, Bernd C.
Mathematics · #11B68 #11B83 #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Mathematical functions and polynomials #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1708.07119

openalex publication_date 2017/08/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let \⋅\ denote the fractional part and n ≥ 1 be a fixed integer. In this short note, we show for any prime p the one-to-one correspondence ∑_{ν≥ 1} \(n)/(p^ν)\ gt; 1 \iff p | denom( B_n(x) - B_n ), where B_n(x) - B_n is the nth Bernoulli polynomial without constant term and denom(⋅) is its denominator, which is squarefree.

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