arXiv subjects

Sergei Konyagin

Publications and source records attributed to Sergei Konyagin.

At least 19 recordsLinked to original sources

arXiv2026

Upper and lower estimates for integer complexity

Let $\|n\|$ stand for the integer complexity of the number $n$, i.e. for the least number of $1$'s needed to write $n$ using arbitrary many additions, multiplications, and parentheses. The two-sided inequality $3\log_3 n\leq\|n\|\leq 3\log_2 n$ for all $n$ is well known and reveals the logarithmic behaviour of the complexity function $\|n\|$. While the lower bound $3\log_3 n$ is attained infinitely many times at powers of $3$, the best upper estimate is still unknown, although there are some improvements of the trivial bound $3\log_2 n$. Besides, for $``$typical$"$ numbers, i.e. for almost all numbers $n$, the better inequality $\|n\|\leq C_{avg}\log n$ holds, where, importantly, $C_{avg}\approx 3.236<\sup_{n} \frac{\|n\|}{\log n}$. We show that in fact $\|n\|\leq C_{avg}\log n+o(\log n)$ as $n\to\infty$, which, in particular, yields that $\limsup\limits_{n\to\infty}\frac{\|n\|}{\log n}\leq C_{avg}$. We also obtain the first nontrivial lower bound $\|n\|\geq 3.06\log_3 n$ for almost all numbers $n$.

math.NT↗

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.

math.NT↗

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.

math.NT↗

arXiv2022

Long gaps in sieved sets

For each prime $p$, let $I_p \subset \mathbb{Z}/p\mathbb{Z}$ denote a collection of residue classes modulo $p$ such that the cardinalities $|I_p|$ are bounded and about $1$ on average. We show that for sufficiently large $x$, the sifted set $\{ n \in \mathbb{Z}: n \pmod{p} \not \in I_p \hbox{ for all }p \leq x\}$ contains gaps of size at least $x (\log x)^δ $ where $δ>0$ depends only on the density of primes for which $I_p\ne \emptyset$. This improves on the "trivial" bound of $\gg x$. As a consequence, for any non-constant polynomial $f:\mathbb{Z}\to \mathbb{Z}$ with positive leading coefficient, the set $\{ n \leq X: f(n) \hbox{ composite}\}$ contains an interval of consecutive integers of length $\ge (\log X) (\log\log X)^δ$ for sufficiently large $X$, where $δ>0$ depends only on the degree of $f$.

math.NT↗

arXiv2021

Sampling discretization of the uniform norm

Discretization of the uniform norm of functions from a given finite dimensional subspace of continuous functions is studied. We pay special attention to the case of trigonometric polynomials with frequencies from an arbitrary finite set with fixed cardinality. We give two different proofs of the fact that for any $N$-dimensional subspace of the space of continuous functions it is sufficient to use $e^{CN}$ sample points for an accurate upper bound for the uniform norm. Previous known results show that one cannot improve on the exponential growth of the number of sampling points for a good discretization theorem in the uniform norm. Also, we prove a general result, which connects the upper bound on the number of sampling points in the discretization theorem for the uniform norm with the best $m$-term bilinear approximation of the Dirichlet kernel associated with the given subspace. We illustrate application of our technique on the example of trigonometric polynomials.

math.NA↗

arXiv2021

A problem in comparative order theory

Write $\mathrm{ord}_p(\cdot)$ for the multiplicative order in $\mathbb{F}_p^{\times}$. Recently, Matthew Just and the second author investigated the problem of classifying pairs $α, β\in \mathbb{Q}^{\times}\setminus\{\pm 1\}$ for which $\mathrm{ord}_p(α) > \mathrm{ord}_p(β)$ holds for infinitely many primes $p$. They called such pairs order-dominant. We describe an easily-checkable sufficient condition for $α,β$ to be order-dominant. Via the large sieve, we show that almost all integer pairs $α,β$ satisfy our condition, with a power savings on the size of the exceptional set.

math.NT↗

arXiv2021

Riesz projection and bounded mean oscillation for Dirichlet series

We prove that the norm of the Riesz projection from $L^\infty(\Bbb{T}^n)$ to $L^p(\Bbb{T}^n)$ is $1$ for all $n\ge 1$ only if $p\le 2$, thus solving a problem posed by Marzo and Seip in 2011. This shows that $H^p(\Bbb{T}^{\infty})$ does not contain the dual space of $H^1(\Bbb{T}^{\infty})$ for any $p>2$. We then note that the dual of $H^1(\Bbb{T}^{\infty})$ contains, via the Bohr lift, the space of Dirichlet series in $\operatorname{BMOA}$ of the right half-plane. We give several conditions showing how this $\operatorname{BMOA}$ space relates to other spaces of Dirichlet series. Finally, relating the partial sum operator for Dirichlet series to Riesz projection on $\Bbb{T}$, we compute its $L^p$ norm when $1<p<\infty$, and we use this result to show that the $L^\infty$ norm of the $N$th partial sum of a bounded Dirichlet series over $d$-smooth numbers is of order $\log\log N$.

math.FA↗

arXiv2020

Gaps between totients

We study the set D of positive integers d for which the equation $ϕ(a)-ϕ(b)=d$ has infinitely many solution pairs (a,b), where $ϕ$ is Euler's totient function. We show that the minumum of D is at most 154, exhibit a specific A so that every multiple of A is in D, and show that any progression a mod d with 4|a and 4|d, contains infinitely many elements of D. We also show that the Generalized Elliott-Halberstam Conjecture, as defined in [6], implies that D equals the set of all positive, even integers.

math.NT↗

arXiv2020

Residue classes free of values of Euler's function

We characterize which residue classes contain infinitely many totients (values of Euler's function) and which do not. We show that the union of all residue classes that are totient-free has asymptotic density 3/4, that is, almost all numbers that are 2 mod 4 are in a residue class that is totient-free. In the other direction, we show the existence of a positive density of odd numbers m, such that for any $s\ge0$ and any even number $a$, the residue class $a\pmod{2^sm}$ contains infinitely many totients.

math.NT↗

arXiv2020

Divisibility of the central binomial coefficient $\binom{2n}{n}$

We show that for every fixed $\ell\in\mathbb{N}$, the set of $n$ with $n^\ell|\binom{2n}{n}$ has a positive asymptotic density $c_\ell$, and we give an asymptotic formula for $c_\ell$ as $\ell\to \infty$. We also show that $\# \{n\le x, (n,\binom{2n}{n})=1 \} \sim cx/\log x$ for some constant $c$. One novelty is a method to capture the effect of large prime factors of integers in general sequences.

math.NT↗

arXiv2019

The prime number race and zeros of Dirichlet L-functions off the critical line

Let $π_{q,a}(x)$ denote the number of primes $\le x$ in the progression $a$ modulo $q$. We study subtle inequities in these functions, with $q$ fixed and variable $a$ (sometimes called 'prime race problems'). It is known unconditionally for many triples $(q,a,b)$ that the difference $π_{q,a}(x) - π_{q,b}(x)$ changes sign infinitely often, although there may be a pronounced bias toward one sign (first observed by Chebyshev in 1853). Similar results for the comparison of three or more prime counting functions all require the assumption of ERH (extended Riemann Hypothesis for the Dirichlet L-functions modulo $q$). In this paper we show that the assumption of ERH is, in a sense, necessary. That is, we prove, for any quadruple $(q,a,b,c)$ with $a,b,c$ co-prime to $q$, that the existence of certain hypothetical configurations of zeros of Dirichlet L-functions lying off the critical line imply that one of the six possible orderings of the three functions $π_{q,a}(x), π_{q,b}(x), π_{q,c}(x)$ does not occur at all for large enough $x$.

math.NT↗

arXiv2019

The prime number race and zeros of Dirichlet L-functions off the critical line, II

We continue our examination the effects of certain hypothetical configurations of zeros of Dirichlet $L$-functions lying off the critical line ("barriers") on the relative magnitude of the functions $π_{q,a}(x)$. Here $π_{q,a}(x)$ is the number of primes $\le x$ in the progression $a \mod q$. In particular, we construct barriers so that $π_{q,1}(x)$ is simultaneously greater than, or simultaneously less than, each of $k$ functions $π_{q,a_i}(x)$ ($1\le i\le k$). We also construct barriers so that only a small number of the $r!$ possible orderings of functions $π_{q,a_i}(x)$ ($1\le i\le r$) occur for large $x$; see Theorem 5.1.

math.NT↗

arXiv2019

Chebyshev's conjecture and the prime number race

We survey results about prime number races, that is, results about the relative sizes of prime counting functions $π_{q,a}(x)$, with $q$ fixed and $a$ varying. In particular, we describe recent work by the authors on these problems.

math.NT↗

arXiv2019

On two conjectures of Sierpiński concerning the arithmetic functions $σ$ and $ϕ$

Let $σ(n)$ denote the sum of the positive divisors of $n$. We prove that for any positive integer $k$, there is a number $m$ for which the equation $σ(x)=m$ has exactly $k$ solutions, settling a conjecture of Sierpiński from 1955. Additionally, it is shown that for every positive even $k$, there is a number $m$ for which the equation $ϕ(x)=m$ has exactly $k$ solutions, where $ϕ$ is Euler's function, making progress toward another conjecture of Sierpiński from 1955.

math.NT↗

arXiv2019

An extremal problem for integer sparse recovery

Motivated by the problem of integer sparse recovery we study the following question. Let $A$ be an $m \times d$ integer matrix whose entries are in absolute value at most $k$. How large can be $d=d(m,k)$ if all $m \times m$ submatrices of $A$ are non-degenerate? We obtain new upper and lower bounds on $d$ and answer a special case of the problem by Brass, Moser and Pach on covering $m$-dimensional $k \times \cdots\times k$ grid by linear subspaces.

math.CO↗

arXiv2017

On subgraphs of random Cayley sum graphs

We prove that asymptotically almost surely, the random Cayley sum graph over a finite abelian group $G$ has edge density close to the expected one on every induced subgraph of size at least $\log^c |G|$, for any fixed $c > 1$ and $|G|$ large enough.

math.CO↗

arXiv2016

Long gaps between primes

Let $p_n$ denotes the $n$-th prime. We prove that $$\max_{p_{n+1} \leq X} (p_{n+1}-p_n) \gg \frac{\log X \log \log X\log\log\log\log X}{\log \log \log X}$$ for sufficiently large $X$, improving upon recent bounds of the first three and fifth authors and of the fourth author. Our main new ingredient is a generalization of a hypergraph covering theorem of Pippenger and Spencer, proven using the Rödl nibble method.

math.NT↗

arXiv2016

New results on sum-products in R

We improve a previous sum--products estimates in R, namely, we obtain that max{|A+A|,|AA|} \gg |A|^{4/3+c}, where c any number less than 5/9813. New lower bounds for sums of sets with small the product set are found. Also we prove some pure energy sum--products results, improving a result of Balog and Wooley, in particular.

math.CO↗