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

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.