This article is cited in 3 papers

INTERNATIONAL CONFERENCE IN MEMORY OF A. A. KARATSUBA ON NUMBER THEORY AND APPLICATIONS

Short Weyl sums and their applications

Z. Kh. Rakhmonov, N. N. Nazrubloev, A. O. Rakhimov

Institute of Mathematics, Academy of Sciences of Republic of Tajikistan, Dushanbe

Abstract: We shall study the behavior of short Weyl sums of the form
$$ T(\alpha ,x,y)=\sum_{x-y<m\leq x}e(\alpha m^n) $$
on major arcs and obtain an asymptotic formula for the number of representations of a sufficiently large positive integer $N$ as a sum of 33 fifth powers of positive integers $x_i$, that satisfy $ \left|x_i-\left(\dfrac{N}{33}\right)^{\frac 15}\right|\le H$, $H\ge N^{\frac 15-\frac{1}{340}+\varepsilon}$.
Bibliography: 17 titles.

Keywords: Short Weyl sums, Almost equal summands, Circle metods, Waring's problem.

UDC: 511.524

Received: 16.02.2015

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