Value distribution of Ramanujan sums and of cyclotomic polynomial coefficients
Pieter Moree, Huib Hommersom
Abstract
The Ramanujan sum cn(k) and an(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 cn(k) (following A. Wintner) and an(k) (partly following H. Moller). In particular we disprove a 1970 conjecture of H. Moller on the average (over n) of an(k). We show that certain symmetric functions in primitive roots considered by the Dence brothers are related to the behaviour of cp-1(k) and ap-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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