Journal of the Ramanujan Mathematical Society

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Zeros of Ramanujan Polynomials


Affiliations
1 Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario K7L 3N6, Canada
2 School of Mathematics and Maxwell Institute for Mathematical Sciences, University of Edinburgh, Edinburgh, Scotland EH9 3JZ, United Kingdom
     

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In this paper, we investigate the properties of Ramanujan polynomials, a family of reciprocal polynomials with real coefficients originating from Ramanujan’s work. We begin by finding their number of real zeros, establishing a bound on their sizes, and determining their limiting values. Next, we prove that all nonreal zeros of Ramanujan polynomials lie on the unit circle, and are asymptotically uniformly distributed there. Finally, for each Ramunujan polynomial, we find all its zeros that are ischolar_mains of unity.
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  • Zeros of Ramanujan Polynomials

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Authors

M. Ram Murty
Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario K7L 3N6, Canada
Chris Smyth
School of Mathematics and Maxwell Institute for Mathematical Sciences, University of Edinburgh, Edinburgh, Scotland EH9 3JZ, United Kingdom
Rob J. Wang
Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario K7L 3N6, Canada

Abstract


In this paper, we investigate the properties of Ramanujan polynomials, a family of reciprocal polynomials with real coefficients originating from Ramanujan’s work. We begin by finding their number of real zeros, establishing a bound on their sizes, and determining their limiting values. Next, we prove that all nonreal zeros of Ramanujan polynomials lie on the unit circle, and are asymptotically uniformly distributed there. Finally, for each Ramunujan polynomial, we find all its zeros that are ischolar_mains of unity.