On a problem of Dobrowolski and Williams and the Polya-Vinogradov inequality

In [2] E. Dobrowolski and K.S. Williams considered a problem of obtaining e stimates for the sum Sigma (a+N)(n=a+1)f(n), for a certain class of functio ns f. One specific application of their result is a new method for estimati ng character sums. In particular, they obtain a form of the Polya-Vinogrado v inequality with the constant 1/(2 log 2). In this note we improve their e stimates and obtain, in particular, a form of the Polya-Vinogradov inequali ty with the constant 1/(3 log 3). A nice feature of our estimate is that it is obtained by a very simple argument.