Zeros of Ramanujan-type Polynomials

Abstract

Ramanujan's notebooks contain many elegant identities and one of the celebrated identities is a formula for $\zeta(2k+1)$. In 1972, Grosswald gave an extension of the Ramanujan's formula for $\zeta(2k+1)$, which contains a polynomial of degree $2k+2$. This polynomial is now well-known as the Ramanujan polynomial$R_{2k+1}(z)$, first studied by Gun, Murty, and Rath. Around the same time, Murty, Smith and Wang proved that all the non-real zeros of $R_{2k+1}(z)$ lie on the unit circle. Recently, Chourasiya, Jamal, and the first author found a new polynomial while obtaining a Ramanujan-type formula for Dirichlet $L$-functions and named it as Ramanujan-type polynomial $R_{2k+1,p}(z)$. In the same paper, they conjectured that all the non-real zeros of $R_{2k+1,p}(z)$ lie on the circle $|z|=1/p$. The main goal of this paper is to present a proof of this conjecture.