Mean-value theorem for non-complete rational trigonometric sums

Abstract

For 2k > 0.5n(n+1)+1 0 ≤ l ≤ 0,5k−w−1,w = [lnn/lnp,] the asymptotic formulas was proved for the number of solutions of the system of congruences

{x1 +···+ xk ≡ y1 +···+ yk (mod p^m)

xⁿ/₁ +···+ xⁿ/_k ≡ yⁿ/₁ +···+ yⁿ/_k (mod p^m)},

where unknowns x1,...,xk,y1,...,yk run values up 1 to pm−l from the complete system residues by modulo pm. The finding formula for 2k ≤ 0.5n(n + 1) + 1 has no the place.

Let be 1 ≤ s < r < ··· < n,s + r +···+ n < 0.5n(n + 1),0 ≤ l ≤ 0,5k −w−1. Then as2 k > s + r +···+ n for the number of the system of congruencies

{x^s/₁ +•••+ xs/k ≡ y^s/₁ +•••+ y^s/_k (mod pm)

x^r/₁ +•••+ x^r/_k ≡ y^r/₁ +•••+ y^r/_k (mod pm)

xⁿ/₁ +•••+ xⁿ/_k ≡ yⁿ/₁ +•••+ yⁿ/_k (mod pm)},

, where unknowns x₁,...,x_k,y₁,...,y_k run values up 1 to p^m−l from the complete system residues by modulo pm, was found the asymptotic formula. This formula has no place as 2k ≤ s + r +···+ n.