An upper bound for the sum $\sum ^ {a+H}_ {n=a+1}f(n)$ for a certain class of functions $f$

Abstract:

For a certain class of functions $f:Z \to C$ an upper bound is obtained for the sum $\sum \nolimits _{n = a + 1}^{a + H} {f\left ( n \right )}$. This bound is used to give a proof of a classical inequality due to Pólya 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 $\sum \nolimits _{x = 1}^H {( ( {R{g^x} + S} )/p} )$, where $g$ is a primitive root of the odd prime $p,1 \leq H \leq p - 1$ and $RS$ is not divisible by $p$.