Huib Hommersom | Searcharxiv

arXiv2008

Value distribution of cyclotomic polynomial coefficients

Let a_n(k) be the kth coefficient of the nth cyclotomic polynomial Phi_n(x). As n ranges over the integers, a_n(k) assumes only finitely many values. For any such value v we determine the density of integers n such that a_n(k)=v. Also we study the average of the a_n(k). We derive analogous results for the kth Taylor coefficient of 1/Phi_n(x) (taken around x=0), the kth coefficient of the nth reciprocal cyclotomic polynomial. We formulate various open problems.

math.NT↗

arXiv2003

Value distribution of Ramanujan sums and of cyclotomic polynomial coefficients

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.

math.NT↗