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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)
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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}
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    • 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
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