Exponential Sums with a Difference

Abstract

Let F(x) be a real function with sufficiently many derivatives existing and satisfying certain non-vanishing conditions for 1⩽x⩽2. The sum S = ∑ h = H 2 H - 1 ∑ m = M + 2 H - 1 2 M - 2 H e ( T F ( m + h M ) - T F ( m - h M ) ) can be estimated by exponential sum methods, giving bounds for the mean square of a general exponential sum and the result ∫ 0 T | ξ ( 1 2 + i t ) | 2 d t = ( log T / 2 π + 2 y - 1 ) T + O ( T 1 / 22 + ∈ ) for the mean square of the Riemann zeta function.