Lower order terms for the one-level densities of symmetric power $L$-functions in the level aspect

TL;DR

This paper analyzes the lower order terms of the one-level densities of symmetric power L-functions in the level aspect, using Chebyshev polynomials to simplify combinatorial complexities.

Contribution

It introduces a novel approach using Chebyshev polynomials to efficiently compute lower order terms in the one-level density of symmetric power L-functions.

Findings

Lower order terms are explicitly calculated.

Chebyshev polynomials reduce combinatorial complexity.

Results enhance understanding of zero distributions.

Abstract

In a previous paper, the authors determined, among other things, the main terms for the one-level densities for low-lying zeros of symmetric power L-functions in the level aspect. In this paper, the lower order terms of these one-level densities are found. The combinatorial difficulties, which should arise in such context, are drastically reduced thanks to Chebyshev polynomials, which are the characters of the irreducible representations of SU(2). %