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International conference on Analytic Number Theory dedicated to 75th anniversary of G. I. Arkhipov and S. M. Voronin
December 16, 2020 14:00–14:30, Moscow, online
 


Approximations of the given real number by the sums of two square roots of integers.

D. V. Goryashin

Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
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Abstract: We consider the inequality
$$ |\sqrt{n}+\sqrt{m}-x|<\Delta $$
where $m$ and $n$ are natural numbers, $x$ is sufficiently large real number and $0<\Delta<\frac12$. In the talk, we prove the formula for the number of solutions of such inequality of the type
$$ J(x,\Delta) = \frac{2}{3}x^{3}\Delta + O(x^{4/3}(\ln{x})^{7/2}), $$
which is asymptotic for $\Delta\gg x^{-5/3}(\ln{x})^{7/2+\varepsilon}$, $\varepsilon>0$.

* Conference identificator: 947 3270 9056 Password: 555834

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