Zapiski Nauchnykh Seminarov POMI
RUS  ENG    JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB  
General information
Latest issue
Archive
Impact factor

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS


Zap. Nauchn. Sem. POMI:
Year:
Volume:
Issue:
Page:
Find




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

Zapiski Nauchnykh Seminarov POMI, 2010, Volume 377, Pages 199–216 (Mi znsl3821)  
This article is cited in 5 scientific papers (total in 5 papers)

Bounds for the cubic Weyl sum

D. R. Heath-Brown

Mathematical Institute, Oxford
Full-text PDF (601 kB) Citations (5)
References:
PDF
HTML
Abstract: Subject to the $abc$-conjecture, we improve the standard Weyl estimate for cubic exponential sums in which the argument is a quadratic irrational. Specifically we show that
$$ \sum_{n\le N}e(\alpha n^3)\ll_{\varepsilon,\alpha}N^{\frac57+\varepsilon} $$
for any $\varepsilon>0$ and any quadratic irrational $\alpha\in\mathbb R-\mathbb Q$. Classically one would have had the (unconditional) exponent $\frac34+\varepsilon$ for such $\alpha$. Bibl. 5 titles.
Key words and phrases: cubic Weyl sum, quadratic irrational, van der Corput's method, upper bound, exponential sum.
Received: 21.05.2010
English version:
Journal of Mathematical Sciences (New York), 2010, Volume 171, Issue 6, Pages 813–823
DOI: https://doi.org/10.1007/s10958-010-0184-7
Bibliographic databases:
Document Type: Article
UDC: 511.33
Language: English
Citation: D. R. Heath-Brown, “Bounds for the cubic Weyl sum”, Studies in number theory. Part 10, Zap. Nauchn. Sem. POMI, 377, POMI, St. Petersburg, 2010, 199–216; J. Math. Sci. (N. Y.), 171:6 (2010), 813–823
Citation in format AMSBIB
\Bibitem{Hea10}
\by D.~R.~Heath-Brown
\paper Bounds for the cubic Weyl sum
\inbook Studies in number theory. Part~10
\serial Zap. Nauchn. Sem. POMI
\yr 2010
\vol 377
\pages 199--216
\publ POMI
\publaddr St.~Petersburg
\mathnet{http://mi.mathnet.ru/znsl3821}
\transl
\jour J. Math. Sci. (N. Y.)
\yr 2010
\vol 171
\issue 6
\pages 813--823
\crossref{https://doi.org/10.1007/s10958-010-0184-7}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-78650057363}
Linking options:
  • https://www.mathnet.ru/eng/znsl3821
  • https://www.mathnet.ru/eng/znsl/v377/p199
    • 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
    		Записки научных семинаров ПОМИ
    Statistics & downloads:
    Abstract page:151
    Full-text PDF :57
    References:37
     
      Contact us:
    		  Terms of Use  Registration to the website  Logotypes © Steklov Mathematical Institute RAS, 2024