Some Numerical Results on Fekete Polynomials

Abstract

It is known that if $\chi$ is a real residue character modulo k with $\chi (p) = - 1$ for the first five primes p, then the corresponding Fekete polynomial $\Sigma _{n = 1}^k\;\chi (n){x^n}$ changes sign on (0, 1). In this paper it is shown that the condition that $\chi (p)$ be -1 for the first four primes p is not sufficient to guarantee such a sign change. More specifically, if $\chi$ is the real nonprincipal character modulo either 1277 or 1973, it is shown that the corresponding Fekete polynomial is positive throughout (0, 1) even though $\chi (2) = \chi (3) = \chi (5) = \chi (7) = - 1$.