Using the method of Bombieri and Iwaniec, new upper bounds are obtained for the absolute value of exponential sums S = H/2<hH M/2<mM exp 2�IiTF m + h M . 2�IiTF m . h M with parameters T and M large, T1/3 <M < T2/3 and 1 H <MT .17/57, and with certain non-vanishing conditions imposed on the derivatives of the function F : [1/3, 3] �¨ R. For E(T), the error term in the asymptotic formula for the mean square of |�A(1/2 + it)|, one consequently obtains a new bound E(T) T�AE log�O(T + 2), with a constant �O < 4, and with the same �emain exponent�f, �AE = 131/416, as occurs in the sharpest upper bounds yet found for the error terms of the circle and divisor problems. The proofs utilize recent advances by Huxley, applying to the Bombieri.Iwaniec method in general, and independent progress of the author on a question specific to the sum S. Conditional improvements are given, subject to Huxley�fs �eHypothesis H(�E, �E)�f concerning the number of integer points near a given bounded curve.