Zeros of Ramanujan-type polynomials

Abstract

Ramanujans notebooks contain many elegant identities and one of the well known identities is a formula for (2k+1). Grosswald [8] gave an extension of the aforementioned formula for (2k+1) which contains a polynomial of degree 2k+2. This polynomial is now known as the Ramanujan polynomial R2k+1(z). Murty, Smith and Wang [10] proved that all the complex zeros of R2k+1(z) lie on the unit circle. Recently, Chourasiya, Jamal, and Maji [5] found a new polynomial while obtaining a Ramanujan-type formula for Dirichlet L-function and named it as Ramanujan-type polynomial R2k+1p(z). In the same paper, they conjectured that all the complex zeros of R2k+1p(z) will lie on the unit circle. One of the main goals of this thesis is to present a proof of their conjecture.