M. R. Gabdullin

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20202023

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math.NT2023

Karatsuba's divisor problem and related questions

Mikhail R. Gabdullin, Vitalii V. Iudelevich, Sergei V. Konyagin

We prove that $p \leq x \sum \frac{1}{τ ( p - 1 )} ≍ \frac{x}{( lo g x ) ^{3/2}}, n \leq x \sum \frac{1}{τ ( n ^{2} + 1 )} ≍ \frac{x}{( lo g x ) ^{1/2}},$ where $τ(n)=\s…

math.NT2022

Prime avoiding numbers is a basis of order $2$

Mikhail R. Gabdullin

For a positive integer $n$ , we denote by $F (n)$ the distance from $n$ to the nearest prime number. We prove that every sufficiently large positive integer $N$ can be represented as…

math.NT2022

A large integer is a sum of two prime avoiding numbers

Artyom Radomskii

Let $f (n) = min_{p} ∣ n - p ∣$ , where $p$ is a prime. We show that there is a positive constant $δ$ such that for any large integer $N$ there exist two positive integers $n_{1}$ and $n_{2}$ …

math.NT2022

Trigonometric polynomials with frequencies in the set of squares

Mikhail R. Gabdullin

Let $γ_{0} = \frac{5 - 1}{2} = 0.618 \dots$ . We prove that, for any $ε > 0$ and any trigonometric polynomial $f$ with frequencies in the set $\{n^2: N \leqslant n\leqslant N+…

math.NT2020

The stochasticity parameter of quadratic residues

Mikhail R. Gabdullin

Following V. I. Arnold, we define the stochasticity parameter $S (U)$ of a subset $U$ of $Z / M Z$ to be the sum of squares of the consecutive distances between eleme…

Karatsuba's divisor problem and related questions

Prime avoiding numbers is a basis of order 2

A large integer is a sum of two prime avoiding numbers

Trigonometric polynomials with frequencies in the set of squares

The stochasticity parameter of quadratic residues

Prime avoiding numbers is a basis of order $2$