Mean value of Weyl sums

G. I. Arkhipov

Matematičeskie zametki, Tome 23 (1978) no. 6, pp. 785-788.

Résumé

We give a simple proof of a mean value theorem of I. M. Vinogradov in the following form. Suppose $P$, $n$, $k$, $\tau$ are integers, $P\ge1$, $n\ge2$, $k\ge n(\tau+1)$, $\tau\ge0$. Put $$ J_{k,n}(P)=\int_0^1\dots\int_0^1\biggl|\sum_{x=1}^Pe^{2\pi i(\alpha_1x+\dots+\alpha_nx^n)}\biggr|^{2k}\,d\alpha_1\dots d\alpha_n. $$ Then $$ J_{k,n}\le n!k^{2n\tau}n^{\sigma n^2u}\cdot2^{2n^2\tau}P^{2k-\Delta}, $$ where \begin{gather*} u=u_\tau=\min(n+1,\tau) \\ \Delta=\Delta_t=n(n+1)/2-(1-1/n)^{\tau+1}n^2/2. \end{gather*}