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Alexander Kalmynin

Publications and source records attributed to Alexander Kalmynin.

12 recordsLinked to original sources
arXiv2025

On additive irreducibility of multiplicative subgroups

In this paper, we employ a version of Stepanov's method, developed by Hanson and Petridis, to prove several results on additive irreducibility of multiplicative subgroups in finite fields of prime order $p$. Specifically, we show that if a subgroup $μ_d$ of $d$-th roots of unity satisfies $A-A=μ_d\cup\{0\}$, then $d=2$ or $6$. Additionally, we resolve the Sárközy's conjecture on quadratic residues: for prime $p$, the set $\mathcal R_p$ of quadratic residues modulo $p$ cannot be represented as $A+B$ for $A,B$ with $\min(|A|,|B|)>1$. More generally, we prove that if the set of $d$-th roots of unity $μ_d$ is represented non-trivially as $A+B$, then the sizes of summands are equal.

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arXiv2024

Sums of squares and sequences of modular forms

Let $h_n(v)$ be the sequence of rational functions with $$ \frac{h_n(v)}{v}-nh_n(v)+(n-1)h_{n-1}(v)-vh_{n-1}'(v)+\frac{v(v(vh_{n-1}(v))')'}{4}=0 $$ for $n>0$ and $h_0(v)=1$. We prove that $h_n(v)$ has a pole at $v=\frac{1}{n}$ if and only if $n$ is a sum of two squares of integers. Moreover, if $r_2(n)=\#\{(a,b)\in \mathbb Z^2: a^2+b^2=n\}$, then we derive the formula $$ \underset{v=1/n}{\mathrm{Res}}h_n(v)=\frac{(-1)^{n-1}r_2(n)}{n16^n}. $$ The results are then generalized to arbitrary modular forms with respect to $Γ(2)$ and as a consequence we obtain a new criterion for Lehmer's conjecture for Ramanujan's $τ$-function.

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arXiv2023

A polynomial analogue of Jacobsthal function

For a polynomial $f(x)\in \mathbb Z[x]$ we study an analogue of Jacobsthal function, defined by the formula \[ j_f(N)=\max_{m}\{\text{For some } x\in \mathbb N \text{ the inequality } (x+f(i),N)>1 \text{ holds for all }i\leq m\}. \] We prove a lower bound \[ j_f(P(y))\gg y(\ln y)^{\ell_f-1}\left(\frac{(\ln\ln y)^2}{\ln\ln\ln y}\right)^{h_f}\left(\frac{\ln y\ln\ln\ln y}{(\ln\ln y)^2}\right)^{M(f)}, \] where $P(y)$ is the product of all primes $p$ below $y$, $\ell_f$ is the number of distinct linear factors of $f(x)$, $h_f$ is the number of distinct non-linear irreducible factors and $M(f)$ is the average size of the maximal preimage of a point under a map $f:\mathbb F_p\to \mathbb F_p$. The quantity $M(f)$ is computed in terms of certain Galois groups.

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arXiv2023

Large gaps between sums of two squareful numbers

Let $M(x)$ be the length of the largest subinterval of $[1,x]$ which does not contain any sums of two squareful numbers. We prove a lower bound \[ M(x)\gg \frac{\ln x}{(\ln\ln x)^2} \] for all $x\geq 3$. The proof relies on properties of random subsets of the prime numbers.

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arXiv2021

Long nonnegative sums of Legendre symbols

For $0\leq α<1$ and prime number $p$ let $L(α,p)$ be the sum of the first $[αp]$ values of Legendre symbol modulo $p$. We study positivity of $L(α,p)$ and prove that for $|α-\frac13|<2\cdot 10^{-6}$ and for rational $α\leq \frac12$ with denominators in the set $\{1,2,3,4,5,6,8,12\}$ the inequality $L(α,p)\geq 0$ holds for majority of primes.

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arXiv2021

Quadratic characters with positive partial sums

Let $\mathcal L^+$ be the set of all primes $p$ for which the sums of $\left(\frac{n}{p}\right)$ over the interval $[1,N]$ are non-negative for all $N$. We prove that the estimate \[ |\mathcal L^+\cap [1,x]|\ll \frac{x}{\ln x(\ln\ln x)^{c-o(1)}} \] holds for $c\approx 0.0368$

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arXiv2019

Orthorecursive expansion of unity

We study the properties of a sequence cn defined by the recursive relation \[\frac{c_0}{n + 1}+\frac{c_1}{n + 2}+\ldots+\frac{c_n}{2n + 1}=0\] for $n>1$ and $c_0=1$. This sequence also has an alternative definition in terms of certain norm minimization in the space $L^2([0, 1])$. We prove estimates on growth order of $c_n$ and the sequence of its partial sums, infinite series identities, connecting $c_n$ with harmonic numbers $H_n$ and also formulate some conjectures based on numerical computations.

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arXiv2018

Intervals between numbers that are sums of two squares

In this paper, we improve the moment estimates for the gaps between numbers that can be represented as a sum of two squares of integers. We consider certain sum of Bessel functions and prove the upper bound for its weighted mean value. This bound provides estimates for the $γ$-th moments of gaps for all $γ\leq 2$.

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arXiv2017

Large values of short character sums

In this paper, we prove that for any $A>0$ there exist infinitely many primes $p$ for which sums of the Legendre symbol modulo $p$ over an interval of length $(\ln p)^A$ can take large values.

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arXiv2017

On Novák numbers

In this work, we obtain some new lower bounds for the number $\mathcal N_B(x)$ of Novák numbers less than or equal to $x$. We also prove, conditionally on Generalized Riemann Hypothesis, the upper estimates for the number of primes dividing at least one Novák number and give description for the prime factors of Novák numbers $N$, such that $2N$ is a Novák-Carmichael number.

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arXiv2017

Novák-Carmichael numbers and shifted primes without large prime factors

We prove some new lower bounds for the counting function $\mathcal N_{\mathcal C}(x)$ of the set of Novák-Carmichael numbers. Our estimates depend on the bounds for the number of shifted primes without large prime factors. In particular, we prove that $\mathcal N_{\mathcal C}(x) \gg x^{0.7039-o(1)}$ unconditionally and that $\mathcal N_{\mathcal C}(x) \gg xe^{-(7+o(1))(\log x)\frac{\log\log\log x}{\log\log x}}$, under some reasonable hypothesis.

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