Value distribution of Ramanujan sums and of cyclotomic polynomial coefficients

2003/07/27 by Pieter Moree, Moree, Pieter, Huib Hommersom +1 · 2 citations
Mathematics · #11N37 #11N60 #Advanced Algebra and Geometry #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11N37 #msc:11N60

paper · pdf · doi:10.48550/arxiv.math/0307352

61 pages, 11 tables, based on M.Sc. thesis (literature study) project of H. Hommersom

arxiv created 2003/07/27 · openalex publication_date 2003/07/27 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The Ramanujan sum c_n(k) and a_n(k), the kth coefficient of the nth cyclotomic polynomial, are completely symmetric expressions in terms of primitive nth roots of unity. For fixed k we study the value distribution of c_n(k) (following A. Wintner) and a_n(k) (partly following H. Moller). In particular we disprove a 1970 conjecture of H. Moller on the average (over n) of a_n(k). We show that certain symmetric functions in primitive roots considered by the Dence brothers are related to the behaviour of c_p-1(k) and a_p-1(k) as p ranges over the primes and study their value distribution as well. This paper is an outgrowth of the M.Sc. thesis project of the second author, carried out under the supervision of the first author at the Korteweg-de Vries Institute (University of Amsterdam) and is written in M.Sc. thesis style. We gratefully acknowledge numerical assistance by Yves Gallot.

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