Variations of the Ramanujan polynomials and remarks on $\zeta(2j+1)/\pi^{2j+1}$

TL;DR

This paper investigates five polynomial families with zeros on the unit circle, proving this property for four, and explores their connections to Ramanujan polynomials involving Bernoulli, Euler numbers, and zeta values.

Contribution

It explicitly proves the zero distribution for four polynomial families related to Ramanujan polynomials, expanding understanding of their properties and connections to special number sequences.

Findings

Five polynomial families have all zeros on the unit circle.

Explicit proofs for four of these polynomial families.

Connections established between these polynomials and special number sequences.

Abstract

We observe that five polynomial families have all of their zeros on the unit circle. We prove the statements explicitly for four of the polynomial families. The polynomials have coefficients which involve Bernoulli numbers, Euler numbers, and the odd values of the Riemann zeta function. These polynomials are closely related to the Ramanujan polynomials, which were recently introduced by Murty, Smyth and Wang. Our proofs rely upon theorems of Schinzel, and Lakatos and Losonczi and some generalizations.