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Mat. Sb. (N.S.), 1983, Volume 122(164), Number 4(12), Pages 527–545 (Mi msb2314)  

This article is cited in 4 scientific papers (total in 4 papers)

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

D. A. Mit'kin


Abstract: Suppose that $n\geqslant2$, $q>1$ and $P\geqslant1$ are integers, $P<q$, $f(x)=a_nx^n+…+a_1x$ is a polynomial with integer coefficients, and $(a_n,…,a_2,q)=d$. 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
$$ |s(f,q,p)|\ll pq^{-\frac1n+\varepsilon}d^\frac1n+q^{1-\frac1n+\varepsilon}d^\frac1n(\frac qd)^{-\rho}, $$
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.

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English version:
Mathematics of the USSR-Sbornik, 1985, 50:2, 513–532

Bibliographic databases:

UDC: 511.3
MSC: Primary 10G10; Secondary 10G05, 10A10
Received: 11.01.1983

Citation: D. A. Mit'kin, “On estimates and asymptotic formulas for rational trigonometric sums that are almost complete”, Mat. Sb. (N.S.), 122(164):4(12) (1983), 527–545; Math. USSR-Sb., 50:2 (1985), 513–532

Citation in format AMSBIB
\Bibitem{Mit83}
\by D.~A.~Mit'kin
\paper On estimates and asymptotic formulas for rational trigonometric sums that are almost complete
\jour Mat. Sb. (N.S.)
\yr 1983
\vol 122(164)
\issue 4(12)
\pages 527--545
\mathnet{http://mi.mathnet.ru/msb2314}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=725456}
\zmath{https://zbmath.org/?q=an:0554.10022|0539.10030}
\transl
\jour Math. USSR-Sb.
\yr 1985
\vol 50
\issue 2
\pages 513--532
\crossref{https://doi.org/10.1070/SM1985v050n02ABEH002842}


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    This publication is cited in the following articles:
    1. Shparlinskii I., “Polynomial Congruences”, Acta Arith., 58:2 (1991), 153–156  crossref  mathscinet  isi
    2. S. V. Konyagin, T. Steger, “On polynomial congruences”, Math. Notes, 55:6 (1994), 596–600  mathnet  crossref  mathscinet  zmath  isi
    3. Shparlinski I., “On Exponential Sums with Sparse Polynomials and Rational Functions”, J. Number Theory, 60:2 (1996), 233–244  crossref  mathscinet  zmath  isi
    4. Cochrane T., Zheng Z., “A Survey on Pure and Mixed Exponential Sums Modulo Prime Powers”, Number Theory for the Millennium I, eds. Bennett M., Berndt B., Boston N., Diamond H., Hildebrand A., Philipp W., A K Peters, Ltd, 2002, 273–300  mathscinet  zmath  isi
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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