An Upper Bound for the Sum ∑ a + H n = a + 1 f(n) for a Certain Class of Functions f

For a certain class of functions f: Z -C an upper bound is obtained for the sum ~a+H I f(n) . This bound is used to give a proof of a classical inequality due to P6lya and Vinogradov that does not require the value of the modulus of the Gauss sum and to obtain an estimate of the sum of Legendre symbols EHj, ((Rgx + S)/p), where g is a primitive root of the odd prime p, 1 < H < p 1 and RS is not divisible by p .