0803.2483Mar 17, 200826 pages, 6 tables. Partly based on arXiv:math.NT/0307352, which is the M.Sc. thesis of H. Hommersom (2003), enriched with research results due to Moree. Some of these research results have been extracted and various new results added. A connection with reciprocal cyclotomic polynomials (arXiv:0709.1570) is also made

Value distribution of cyclotomic polynomial coefficients

Yves Gallot, Pieter Moree, Huib Hommersom

Number Theory

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Abstract

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.

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