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Conference in memory of A. A. Karatsuba on number theory and applications
January 31, 2014 15:40–15:55, Moscow, Steklov Mathematical Institute, Lecture Room 530

Additive problems with the summands of a special type

D. V. Goryashin
 Video records: Flash Video 155.4 Mb Flash Video 930.7 Mb MP4 155.4 Mb

Abstract: The talk is devoted to the following additive problem. Suppose that $\alpha>1$ is a fixed irrational number. Let $r_3(\alpha,N)$ equals to the number of partitions of $N$ into a sum of two square -free summands and the term of the type $[\alpha q]$ with square -free $q$. In other words, $r_3(\alpha,N)$ is the number of representation $q_1+q_2+[\alpha q_3]=N$ where the numbers $q_1,q_2,q_3$ are square -free. Then the following asymptotic formula holds
$$r_{3}(\alpha,N) = \frac{1}{2\alpha}(\frac{6}{\pi^2})^{3}N^{2}+O(N^{11/6+\varepsilon})$$
as $N\to\infty$.