Extremal values for the sum

Let q be an odd integer, let τ be the order of 2 modulo q, and let a be coprime with q. Finally, let . We prove that |s(a/q)| can be as large as τ−c′ for a suitable constant c′ for infinitely many q, but that max(a,q)=1|s(a/q)| is always bounded from above by τ−c for a suitable positive constant c whose value is considerably larger that any previously known. An upper bound for min(a,q)=1|s(a/q)| is also discussed.