On a Problem of H. Cohn for Character Sums Original Research Article

Cohnʹs problem on character sums (see [4], p. 202) asks whether a multiplicative character on a finite field can be characterized by a kind of two level autocorrelation property. Let f be a map from a finite field F to the complex plane such that f(0)=0, f(1)=1, and f(α)=1 for all α≠0. In this paper we show that if for all a, bset membership, variantF*, we haveimage then f is a multiplicative character of F. We also prove that if F is a prime field and f is a real valued function on F with f(0)=0, f(1)=1, and f(α)=1 for all α≠0, then ∑αset membership, variantF f(α) f(α+a)=−1 for all a≠0 if and only if f is the Legendre symbol. These results partially answer Cohnʹs problem.