On the exponents of the convergence of singular integrals and singular series of a multivariate problem
In the paper we continue studies on the theory of multivariate trigonometric sums, in the base of which lies of the I.M.Vinogradov’s method. Here we obtain for = = 2 lower estimates of the convergence exponent of the singular series and the singular integral of the asymptotic formulas for → ∞for the number of solutions of the following system of Diophantine equations Σ︁2𝑘 𝑗=1 (-1)𝑗𝑥𝑡1 1,𝑗 . . . 𝑟,𝑗 = 0, 0 ≤ 𝑡1, . . . , ≤ 𝑛, where ≥ 2, ≥ 1, are natural numbers, moreover an each variable 𝑥𝑖,𝑗 can take all integer values from 1 to ≥ 1.