On Berndt-Straub conjectures about zeros of generalized Ramanujan polynomials
Abstract
We settle two conjectures of Berndt and Straub asserting that all nonzero zeros of the generalized Ramanujan polynomials Rk(x;χ,ψ) and S2n(x;χ,χ) lie on the unit circle for real nonprincipal Dirichlet characters. We prove their conjecture for Rk. For S2n, we prove that at most four nonzero zeros lie off the unit circle, all purely imaginary. If χ163=(·/163) is the primitive quadratic character modulo $163$, then S6(x;χ163,χ163) has four exceptional zeros, showing that this bound is sharp. This disproves the second conjecture and establishes that the `Riemann hypothesis for period polynomials' can fail for Eisenstein newforms. Further, we introduce a transcendental invariant whose sign determines whether exceptional zeros of S2n(x;χ,χ) occur in all sufficiently large degrees, and show that these zeros converge to transcendental limits off the unit circle. For primitive odd characters, we relate this invariant and these limits to Fekete polynomials.
Related
- On the special values of Berndt's generalized L-function
- Zeros of Ramanujan-Like Polynomials
- Generalized Glaisher-Kinkelin constants and Ramanujan summation of series
- On Igusa's conjectures for weighted homogeneous polynomials with isolated singularities
- The Erdélyi--Magnus--Nevai and Krasikov Conjectures for Jacobi Polynomials
BibTeX
@misc{pathak2026berndtstraub,
title = {{On Berndt-Straub conjectures about zeros of generalized Ramanujan polynomials}},
author = {Siddhi Pathak},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.18065},
eprint = {2610.18065},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.18065},
}
RIS
TY - UNPB
TI - On Berndt-Straub conjectures about zeros of generalized Ramanujan polynomials
AU - Siddhi Pathak
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.18065
UR - https://arxiv.org/abs/2610.18065
AB - We settle two conjectures of Berndt and Straub asserting that all nonzero zeros of the generalized Ramanujan polynomials R_k(x;χ,ψ) and S_2n(x;χ,χ) lie on the unit circle for real nonprincipal Dirichlet characters. We prove their conjecture for R_k. For S_2n, we prove that at most four nonzero zeros lie off the unit circle, all purely imaginary. If χ_163=(·/163) is the primitive quadratic character modulo $163$, then S_6(x;χ_163,χ_163) has four exceptional zeros, showing that this bound is sharp. This disproves the second conjecture and establishes that the `Riemann hypothesis for period polynomials' can fail for Eisenstein newforms. Further, we introduce a transcendental invariant whose sign determines whether exceptional zeros of S_2n(x;χ,χ) occur in all sufficiently large degrees, and show that these zeros converge to transcendental limits off the unit circle. For primitive odd characters, we relate this invariant and these limits to Fekete polynomials.
ER -