On a function introduced by Erd\"{o}s and Nicolas

TL;DR

This paper proves that a specific arithmetical function related to divisors is the largest coefficient of a polynomial and links the polynomial's coefficients to Pythagorean triangles, improving previous results.

Contribution

It establishes the connection between the function's maximum coefficient and Pythagorean triangles, and refines earlier findings on polynomial coefficients.

Findings

F(n) is the largest coefficient of P_n(q)

P_n(q) has a coefficient > 1 iff 2n is a Pythagorean triangle perimeter

Improves a previous result by Vatne on polynomial coefficients

Abstract

Erd\"os and Nicolas [erdos1976methodes] introduced an arithmetical function $F(n)$ related to divisors of $n$ in short intervals $\left] \frac{t}{2}, t\right]$. The aim of this note is to prove that $F(n)$ is the largest coefficient of polynomial $P_n(q)$ introduced by Kassel and Reutenauer [kassel2015counting]. We deduce that $P_n(q)$ has a coefficient larger than $1$ if and only if $2n$ is the perimeter of a Pythagorean triangle. We improve a result due to Vatne [vatne2017sequence] concerning the coefficients of $P_n(q)$.