Fekete polynomials of principal Dirichlet characters

2023/07/27 by Shiva Chidambaram, Ján Mináč, Chidambaram, Shiva +5 · 2 citations
Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2307.14896

openalex publication_date 2023/07/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Fekete polynomials associated to quadratic Dirichlet characters have interesting arithmetic properties, and have been studied in many works. In this paper, we study a seemingly simpler yet rich variant: the Fekete polynomial F_n(x) = ∑_a=1ⁿ χ_n(a) x^a associated to a principal Dirichlet character χ_n of modulus n. We investigate the cyclotomic factors of F_n and conjecturally describe all of them. One interesting observation from our computations is that the non-cyclotomic part f_n of F_n(x)/x seems to be always irreducible. We study this factor closely in the special case that n is a product of two odd primes, proving separability in specific cases, and studying its coefficients and special values. Combining these theoretical results with computational evidence lets us identify the Galois group of f_n for small n, and raises precise questions in general.

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• Modular Periodicity of Random Initialized Recurrences
• Supercharacters of finite abelian groups and applications to spectra of U-unitary Cayley graphs

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