On a character sum problem of H. Cohn

Abstract

Let $f$ be a complex valued function on a finite field $F$ such that $f(0)=0$, $f(1)=1$, and $|f(x)|=1$ for $x\ne 0$. Cohn asked if it follows that $f$ is a nontrivial multiplicative character provided that ${\sum }_{x\in F}f(x)\overline{f(x+h)}=-1$ for $h\ne 0$. We prove that this is the case for finite fields of prime cardinality under the assumption that the nonzero values of $f$ are roots of unity.