Counter-examples to a problem of Cohn on classifying characters (Q1587417)

For a nontrivial multiplicative character \(f(b),(b\in \mathbb{F}_q, q=p^s)\), \(\sum f(b)\overline {f(b+a)}= q-1\) for \(a=0\) and \(-1\) otherwise. The reviewer asked if conversely this sum property implies that \(f(b)\) is a multiplicative character [see \textit{H. L. Montgomery}, Ten lectures on the interface between analytic number theory and harmonic analysis, Regional Conference Series in Mathematics 84, Providence, RI: American Mathematical Society (1994; Zbl 0814.11001)]. \textit{A. Biró} [J. Number Theory 77, 200--208 (1999; Zbl 0949.11039)] proved partial results for \(p=q\) odd, e.g., if \(f(b)=\pm 1\), \(b\neq 0\) then \(f\) is a Legendre character. The authors disprove it emphatically for all other cases (except \(q=2,4)\), using counterexamples where \(|f(b) |=1\) or 0.