Video library: D. V. Goryashin, Additive problems with the summands of a special type

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

Additive problems with the summands of a special type D. V. Goryashin
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 $$ r_{3}(\alpha,N)\,=\,\frac{1}{2\alpha}\biggr(\frac{6}{\pi^2}\biggl)^{\!3}N^{2}+O\bigl(N^{11/6+\varepsilon}\bigr) $$ as $N\to\infty$.