A new kth derivative estimate for exponential sums via Vinogradov's mean value

作者：Heath Brown D R^*

来源：Proceedings of the Steklov Institute of Mathematics, 2017, 296(1): 88-103.

DOI：10.1134/S0081543817010072

We give a slight refinement to the process by which estimates for exponential sums are extracted from bounds for Vinogradov's mean value. Coupling this with the recent works of Wooley, and of Bourgain, Demeter and Guth, providing optimal bounds for the Vinogradov mean value, we produce a powerful new kth derivative estimate. Roughly speaking, this improves the van der Corput estimate for k >= 4. Various corollaries are given, showing for example that zeta(sigma + it) <<(epsilon) t ((1-sigma)3/2) /2+epsilon for t >= 2 and 0 <= sigma <= 1, for any fixed epsilon > 0.

• 出版日期2017-1