arXiv subjects

Mikhail R. Gabdullin

Publications and source records attributed to Mikhail R. Gabdullin.

13 recordsLinked to original sources

arXiv2026

Distinct exponents in the prime factorization

Following Erdős (1982) and Sanna (2019), we study the arithmetic function $h(n)$, which is defined to be the number of distinct exponents in the prime factorization of a positive integer $n$. Among other things, we show that $$ \sum_{n\leq x}h(ϕ(n)) \asymp x\left(\frac{\log\log x}{\log\log\log x}\right)^{1/2}, $$ where $ϕ$ is the Euler totient function. The key ingredient is the Poisson random model for $ω(n,T)$, the number of the prime divisors of $n$ in a given subset of primes $T$, which was introduced in a recent work of Ford.

math.NT↗

arXiv2025

Moments of the shifted prime divisor function

Let $ω^*(n) = \{d|n: d=p-1, \mbox{$p$ is a prime}\}$. We show that, for each integer $k\geq2$, $$ \sum_{n\leq x}ω^*(n)^k \asymp x(\log x)^{2^k-k-1}, $$ where the implied constant may depend on $k$ only. This confirms a recent conjecture of Fan and Pomerance. Our proof uses a combinatorial identity for the least common multiple, viewed as a multiplicative analogue of the inclusion-exclusion principle, along with analytic tools from number theory.

math.NT↗

arXiv2024

Primes with small primitive roots

Let $δ(p)$ tend to zero arbitrarily slowly as $p\to\infty$. We exhibit an explicit set $\mathcal{S}$ of primes $p$, defined in terms of simple functions of the prime factors of $p-1$, for which the least primitive root of $p$ is $\le p^{1/4-δ(p)}$ for all $p\in \mathcal{S}$, where $\#\{p\leq x: p\in \mathcal{S}\} \sim π(x)$ as $x\to\infty$.

math.NT↗

arXiv2024

Trigonometric polynomials with frequencies in the set of cubes

We prove that for any $ε>0$ and any trigonometric polynomial $f$ with frequencies in the set $\{n^3: N \leq n\leq N+N^{2/3-ε}\}$, one has $$ \|f\|_4 \ll ε^{-1/4}\|f\|_2 $$ with implied constant being absolute. We also show that the set $\{n^3: N\leq n\leq N+(0.5N)^{1/2}\}$ is a Sidon set.

math.CA↗

arXiv2023

Trigonometric polynomials with frequencies in the set of squares

Let $γ_0=\frac{\sqrt5-1}{2}=0.618\ldots$ . We prove that, for any $\varepsilon>0$ and any trigonometric polynomial $f$ with frequencies in the set $\{n^2: N \leqslant n\leqslant N+N^{γ_0-\varepsilon}\}$, the inequality $$ \|f\|_4 \ll \varepsilon^{-1/4}\|f\|_2 $$ holds, which makes a progress on a conjecture of Cilleruelo and Cordoba. We also present a connection between this conjecture and the conjecture of Ruzsa which asserts that, for any $\varepsilon>0$, there is $C(\varepsilon)>0$ such that each positive integer $N$ has at most $C(\varepsilon)$ divisors in the interval $[N^{1/2}, N^{1/2}+N^{1/2-\varepsilon}]$

math.NT↗

arXiv2023

Long strings of consecutive composite values of polynomials

We show that for any polynomial $f$ from the integers to the integers, with positive leading coefficient and irreducible over the rationals, if $x$ is large enough then there is a string of $(\log x)(\log\log x)^{1/835}$ consecutive integers $n \in [1,x]$ for which $f(n)$ is composite. This improves a result of the first author, Konyagin, Maynard, Pomerance and Tao, which states that there are such strings of length $(\log x)(\log\log x)^{c_f}$, where $c_f$ depends on $f$ and $c_f$ is exponentially small in the degree of $f$ for some polynomials.

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arXiv2023

Numbers of the form $k+f(k)$

For a function $f\colon \mathbb{N}\to\mathbb{N}$, let $$ N^+_f(x)=\{n\leq x: n=k+f(k) \mbox{ for some } k\}. $$ Let $τ(n)=\sum_{d|n}1$ be the divisor function, $ω(n)=\sum_{p|n}1$ be the prime divisor function, and $φ(n)=\#\{1\leq k\leq n: \gcd(k,n)=1 \}$ be Euler's totient function. We show that \begin{align*} &(1) \quad x \ll N^+_ω(x), \\ &(2) \quad x\ll N^+_τ(x) \leq 0.94x, \\ &(3) \quad x \ll N^+_φ(x) \leq 0.93x. \end{align*}

math.NT↗

arXiv2023

Karatsuba's divisor problem and related questions

We prove that $$ \sum_{p \leq x} \frac{1}{τ(p-1)} \asymp \frac{x}{(\log x)^{3/2}}, \quad \quad \sum_{n \leq x} \frac{1}{τ(n^2+1)} \asymp \frac{x}{(\log x)^{1/2}}, $$ where $τ(n)=\sum_{d|n}1$ is the number of divisors of $n$, and the summation in the first sum is over primes.

math.NT↗

arXiv2022

The stochasticity parameter of quadratic residues

Following V. I. Arnold, we define the stochasticity parameter $S(U)$ of a subset $U$ of $\mathbb{Z}/M\mathbb{Z}$ to be the sum of squares of the consecutive distances between elements of $U$. In this paper we study the stochasticity parameter of the set $R_M$ of quadratic residues modulo $M$. We present a method which allows to find the asymptotics of $S(R_M)$ for a set of $M$ of positive density. In particular, we obtain the following two corollaries. Denote by $s(k)=s(k,\mathbb{Z}/M\mathbb{Z})$ the average value of $S(U)$ over all subsets $U\subseteq \mathbb{Z}/M\mathbb{Z}$ of size $k$, which can be thought of as the stochasticity parameter of a random set of size $k$. Let $\mathfrak{S}(R_M)=S(R_M)/s(|R_M|)$. We show that a) $\varliminf_{M\to\infty}\mathfrak{S}(R_M)<1<\varlimsup_{M\to\infty}\mathfrak{S}(R_M)$; b) the set $\{ M\in \mathbb{N}: \mathfrak{S}(R_M)<1 \}$ has positive lower density.

math.NT↗

arXiv2022

Numbers of the form $kf(k)$

For a function $f\colon \mathbb{N}\to\mathbb{N}$, define $N^{\times}_{f}(x)=\#\{n\leq x: n=kf(k) \mbox{ for some $k$} \}$. Let $τ(n)=\sum_{d|n}1$ be the divisor function, $ω(n)=\sum_{p|n}1$ be the prime divisor function, and $φ(n)=\#\{1\leq k\leq n: (k,n)=1 \}$ be Euler's totient function. We prove that \begin{gather*} \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! 1) \quad N^{\times}_τ(x) \asymp \frac{x}{(\log x)^{1/2}}; \\ 2) \quad N^{\times}_ω(x) = (1+o(1))\frac{x}{\log\log x}; \\ \!\!\!\!\!\!\!\!\! 3) \quad N^{\times}_φ(x) = (c_0+o(1))x^{1/2}, \end{gather*} where $c_0=1.365...$\,.

math.NT↗

arXiv2022

Prime avoiding numbers is a basis of order $2$

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 the sum $N=n_1+n_2$, where $$ F(n_i) \geqslant (\log N)(\log\log N)^{1/325565}, $$ for $i=1,2$. This improves the corresponding "trivial" statement where only $F(n_i)\gg \log N$ is required.

math.NT↗

arXiv2021

Trigonometric series with noninteger harmonics

Let $\{c_k\}$ be a nonincreasing sequence of positive numbers (more general classes of sequences are also considered), and $α>0$ be not an integer. We find necessary and sufficient conditions for the uniform convergence of the series $\sum_k c_k\sin k^αx$ and $\sum_k c_k\cos k^αx$ on the real line and its bounded subsets.

math.CA↗

arXiv2021

Sets whose differences avoid squares modulo m

We prove that if $\varepsilon(m)\to 0$ arbitrarily slowly, then for almost all $m$ and any $A\subset\mathbb{Z}_m$ such that $A-A$ does not contain non-zero quadratic residues we have $|A|\leq m^{1/2-\varepsilon(m)}.$

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