Mathematics of the USSR-Sbornik
RUS  ENG    JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB  
General information
Latest issue
Forthcoming papers
Archive
Impact factor
Guidelines for authors
License agreement
Submit a manuscript

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Mat. Sb.:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


Mathematics of the USSR-Sbornik, 1985, Volume 50, Issue 2, Pages 513–532
DOI: https://doi.org/10.1070/SM1985v050n02ABEH002842
(Mi sm2314)
 

This article is cited in 5 scientific papers (total in 5 papers)

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

D. A. Mit'kin
References:
Abstract: Suppose that $n\geqslant2$, $q>1$ and $P\geqslant1$ are integers, $P<q$, $f(x)=a_nx^n+\dots+a_1x$ 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.
Received: 11.01.1983
Russian version:
Matematicheskii Sbornik. Novaya Seriya, 1983, Volume 122(164), Number 4(12), Pages 527–545
Bibliographic databases:
UDC: 511.3
MSC: Primary 10G10; Secondary 10G05, 10A10
Language: English
Original paper language: Russian
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/sm2314}
\mathscinet{http://mathscinet.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}
Linking options:
  • https://www.mathnet.ru/eng/sm2314
  • https://doi.org/10.1070/SM1985v050n02ABEH002842
  • https://www.mathnet.ru/eng/sm/v164/i4/p527
  • This publication is cited in the following 5 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics
    Statistics & downloads:
    Abstract page:364
    Russian version PDF:91
    English version PDF:16
    References:43
     
      Contact us:
     Terms of Use  Registration to the website  Logotypes © Steklov Mathematical Institute RAS, 2024