A mean value theorem for exponential sums

Abstract

The exponential sum S(x) = ^2e(f{m + x)) has mean square size O(M), when m runs through M consecutive integers, f(x) satisfies bounds on the second and third derivatives, and x runs from 0 to 1. S = y^ e{f{m))(where e(x) = exp2nix) has order of magnitude 5 = O(M l/2 T e ) for any € > 0; the order of magnitude constant may depend on B and on e. It is well-known that 5 has root mean square size \f~M, in the sense that for a > 1, provided that \f'(x)\ > T/BM; the error term may be improved to 0(M 2 /aT) using a theorem of Montgomery and Vaughan [ 1 ]. In this note we prove another mean value