On estimates and asymptotic formulas for rational trigonometric sums that are almost complete

D. A. Mit'kin

Sbornik. Mathematics, Tome 50 (1985) no. 2, pp. 513-532

Résumé

Suppose that $n\geqslant2$, $q>1$ and $P\geqslant1$ are integers, $P$, $f(x)=a_nx^n+\dots+a_1x$ is a polynomial with integer coefficients, and $(a_n,\dots,a_2,q)=d$. Hua proved that an incomplete trigonometric sum of the form $$ s(f,q,p)=\sum_{x=1}^pe^{2\pi i\frac{f(x)}q} $$ satisfies the estimate $$ |s(f,q,p)|\ll q^{1-\frac1n+\varepsilon}d^\frac1n\qquad(\varepsilon>0). $$ In this paper sharper estimates are obtained for $n>2$: $$ |s(f,q,p)|\ll q^{1-\frac1n}d^\frac1n $$ and $$ |s(f,q,p)|\ll pq^{-\frac1n+\varepsilon}d^\frac1n+q^{1-\frac1n+\varepsilon}d^\frac1n\biggl(\frac qd\biggr)^{-\rho}, $$ where $\rho=(n-1)/n(n^2-n+1)$. A consequence of the last estimate is that the same type of estimate holds for the number of solutions of the congruence $$ f(x)\equiv c\pmod q;\qquad1\leqslant x\leqslant p. $$ The proofs are based on estimates for complete rational trigonometric sums with prime power denominator which are obtained by Hua's method (this method has also been developed by V. I. Nechaev, C. Chen, S. B. Stechkin and S. V. Konyagin). Bibliography: 24 titles.