Some numerical results on Fekete polynomials

Abstract

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