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Мemorial conference dedicated to the memory of Ivan Matveevich Vinogradov
March 28, 2019 15:00–15:25, Moscow, Steklov Mathematical Institute, Conference hall
 


Multiplicative graphs ant its applications to the equation $n - \varphi(n)=c$

A. S. Semchankau

Moscow Institute of Physics and Technology (State University), Dolgoprudny, Moscow region
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A. S. Semchankau
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Abstract: We study the following general problem. Given natural number $c$ and a pair of multiplicative functions $f,g$. The question is to find the number of solutions of the equation
$$ f(n) - g(n) = c. $$
Under some conditions to the solutions of this equation and to the functions $f,g$ (in particular, $f(n) > g(n)$ for $n > 1$), we prove that the number of such solutions does not exceed $c^{ 1 - \varepsilon}$. Next, for the number $J(c)$ of solutions of the equation
$$ n - \varphi(n) = c, $$
we find the followingformula:
$$ J(c) = G(c + 1) + O(c^{ 3/4 + o(1)}). $$
Here $G(k)$ denotes the number of representations of $k$ by the sum of two primes. In order to obtain such results, we use so-called multiplicative graphs.

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