A New $k$-th Derivative Estimate for Exponential Sums via Vinogradov's Mean Value

Abstract

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 $k$-th derivative estimate. Roughly speaking, this improves the van der Corput estimate for $k\ge 4$. Various corollaries are given, showing for example that $\zeta (\sigma +it){\ll }_{\epsilon }{t}^{(1-\sigma {)}^{3/2}/2+\epsilon }$ for $t\ge 2$ and $0\le \sigma \le 1$, for any fixed $\epsilon >0$.

Misprints corrected; References to work of Robert added

References in corpus (2)

• The cubic case of the main conjecture in Vinogradov's mean value theorem
• Proof of the main conjecture in Vinogradov's mean value theorem for degrees higher than three

Cited by in corpus (1)

• Hua's Theorem with the Primes in Piatetski-Shapiro Prime Sets