A note on the mean square of vertical bar zeta(1/2+it)vertical bar

作者：Watt N^*

来源：Journal of the London Mathematical Society-Second Series, 2010, 82: 279-294.

DOI：10.1112/jlms/jdq024

Using the method of Bombieri and Iwaniec, new upper bounds are obtained for the absolute value of exponential sums with parameters T and M large, T(1/3) < M < T(2/3) and 1 <= H < MT(-17/57), and with certain non-vanishing conditions imposed on the derivatives of the function F : [1/3, 3]. For E(T), the error term in the asymptotic formula for the mean square of vertical bar zeta(1/2 + it)vertical bar, one consequently obtains a new bound E(T) < T(theta)log(phi) (T + 2), with a constant phi < 4, and with the same 'main exponent', theta = 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's 'Hypothesis H(kappa, lambda)' concerning the number of integer points near a given bounded curve.

• 出版日期2010-10