Explicit van der Corput's $d$-th derivative estimate

TL;DR

This paper provides an explicit, corrected version of van der Corput's $d$-th derivative estimate for exponential sums, with precise constants, improving the accuracy and applicability of the classical method.

Contribution

The paper corrects an error in van der Corput's original 1937 proofs and derives explicit constants for the derivative estimate, enhancing its precision and utility.

Findings

Derived explicit bounds with concrete constants for exponential sums.

Corrected historical errors in van der Corput's original proofs.

Applied the theorem to zeta sums and related it to Titchmarsh's theorem.

Abstract

We give an explicit version for van der Corput's $d$-th derivative estimate of exponential sums. $ \textbf{Theorem.}$ Let $X$, and $Y\in\mathbb{R}$ be such that $\lfloor Y\rfloor>d$ where $d\ge3$ is a natural number. Let $f\colon(X,X+Y]\to\mathbb{R}$ be a real function with continuous derivatives up to the order $d$. Assume that $0<\lambda\le f^{(d)}(x)\le\Lambda$ for $X<x\le X+Y$. Denote by $D=2^d$. Then \begin{equation}\Bigl|\frac{1}{Y}\sum_{X<n\le X+Y}e(f(n))\Bigr|\le\max\Bigl\{A_d\Bigl(\frac{\Lambda}{\lambda Y}\Bigr)^{2/D}, B_d\Bigl(\frac{\Lambda^2}{\lambda}\Bigr)^{1/(D-2)},C_d(\lambda Y^d)^{-2/D}\Bigr\},\end{equation} where $A_d$, $B_d$, and $C_d$ are explicit constants. They depend on $d$ but for $d\ge2$ for example $A_d< 7.5$, $B_d<5.8$ and $C_d<10.9$. We follow the reasoning of van der Corput in three papers published in 1937, that contained an error. I correct this error and try to get the smallest possible constants. We apply this theorem to zeta sums, giving the best choice of $d$ in each case. Also, we prove that our Theorem implies Titchmarsh's Theorem 5.13.