Explicit van der Corput's d-th derivative estimate

We give an explicit version for van der Corput's d-th derivative estimate of exponential sums. Theorem. Let X, and Y be such that Y>d where d≥3 is a natural number. Let f(X,X+Y] be a real function with continuous derivatives up to the order d. Assume that 0<λ≤ f⁽ᵈ⁾(x)≤Λ for X<x≤ X+Y. Denote by D=2ᵈ. Then {equation}|1/Y∑X<n≤ X+Ye(f(n))|≤max\{A_d(Λ/λ Y)2/D, B_d(Λ^2/λ)1/(D-2),C_d(λ Y^d)-2/D\},{equation} where Ad, Bd, and Cd are explicit constants. They depend on d but for d≥2 for example Ad< 7.5, Bd<5.8 and Cd<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.