Asymptotic formula for the mean value of a multiple trigonometric sum

Abstract

When k≥k₀=10 Mr²n log (rn) we have for the trigonometric integral

$$J_n (k,P) = \int_E {|S(A)|^{2k} dA,} $$

where

$$\begin{gathered} S(A) = \sum _{x_1 = 1}^P \cdots \sum _{x_r = 1}^P \exp (2\pi if_A (x_1 , \ldots ,x_r )), \hfill \\ f_A (x_1 , \ldots ,x_r ) = \sum _{t_1 = 0}^n \cdots \sum _{t_r = 0}^n \alpha _{t_1 \cdots l_r } x_1^{t_1 } \cdots x_{r^r }^t \hfill \\ \end{gathered} $$

and E is the M-dimensional unit cube, the asymptotic formula

$$J_n (k,P) = \sigma \theta P^{2kr - rnM/2} + O(P^{2kr - rnM/2 - 1/(2M)} ) + O(P^{2kr - rnM/2 - 1/(500r^2 \log (rn))} ),$$

where σ is a singular series and θ is a singular integral.