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Oleksiy Klurman

Publications and source records attributed to Oleksiy Klurman.

At least 19 recordsLinked to original sources
arXiv2026

On the generation of multiplicative groups by small primes

Motivated by a question of Regev arising from his improved quantum factoring algorithm, we study how many small primes are needed to generate the group $({\mathbb Z}/q{\mathbb Z})^\times$ when each prime may be used with exponent only $0$ or $1$. We prove that, for every fixed $\varepsilon>0$ and $A>0$, there is an absolute constant $C_*$ and a set of at most $(\log Q)^{1+\varepsilon}$ primes, all at most $(\log Q)^{C_*(A+1)}$, such that for all but $O(Q(\log Q)^{-A})$ (with the implied constant depending only on $\varepsilon$ and $A$) integers $q\leq Q$, every element of $({\mathbb Z}/q{\mathbb Z})^\times$ is a product of a subset of these primes modulo $q$. The exponent $1+\varepsilon$ in the number of primes is best possible up to the arbitrary $\varepsilon$ in the exponent.

math.NT↗
arXiv2026

Partition regularity of generalized Pythagorean pairs

We address partition regularity problems for homogeneous quadratic equations. A consequence of our main results is that, under natural conditions on the coefficients $a,b,c$, for any finite coloring of the positive integers, there exists a solution to $ax^2+by^2=cz^2$ where $x$ and $y$ have the same color (and similar results for $x,z$ and $y,z$). For certain choices of $(a,b,c)$, our result is conditional on an Elliott-type conjecture. Our proofs build on and extend previous arguments of the authors dealing with the Pythagorean equation. We make use of new uniformity properties of aperiodic multiplicative functions and concentration estimates for multiplicative functions along arbitrary binary quadratic forms.

math.CO↗
arXiv2026

Partition regularity of Pythagorean pairs

We address a core partition regularity problem in Ramsey theory by proving that every finite coloring of the positive integers contains monochromatic Pythagorean pairs, i.e., $x,y\in \mathbb{N}$ such that $x^2\pm y^2=z^2$ for some $z\in \mathbb{N}$. We also show that partitions generated by level sets of multiplicative functions taking finitely many values always contain Pythagorean triples. Our proofs combine known Gowers uniformity properties of aperiodic multiplicative functions with a novel and rather flexible approach based on concentration estimates of multiplicative functions.

math.CO↗
arXiv2026

Small values of signed harmonic sums and logarithmic means of multiplicative functions

We construct sequences $\{a_n\}_{n\in\mathbb{N}}\in\{-1,1\}^{\mathbb{N}}$ with small values of signed harmonic sums \[ \sum_{n\in\mathcal{A}\cap[1,N]}\frac{a_n}{n}, \] for any reasonably dense subsets $\mathcal{A}\subset\mathbb{N}.$ We apply these methods to further construct completely multiplicative functions $f:\mathbb{N}\to\{-1,1\}$ with unusually small logarithmic partial sums, that is, \[ \sum_{n \leq N}\frac{f(n)}{n} \ll \exp\left(-c_0 \frac{N^{1/3}}{(\log N)^{1/3}} \right) \] holds for infinitely many $N\to\infty$. The proofs combine careful analysis of the small-scale distribution of random harmonic sums over subsets of $\mathbb{N}$, together with deterministic inductive arguments inspired by the ``anatomy" of integers.

math.NT↗
arXiv2026

A Halász-type asymptotic formula for logarithmic means and its consequences

We establish an asymptotic formula for the logarithmic mean value of a 1-bounded multiplicative function that is sharp in many cases of interest. We derive from it a variety of applications, making progress on several old problems. As a first application, we show that if $f$ is a completely multiplicative function taking values in $[-1,1]$ then there is a constant $c > 0$ such that for every $x \geq 3$, $$ L_f(x) := \sum_{n \leq x} \frac{f(n)}{n} > -\frac{c}{(\log x)^{1-2/π}}, $$ thus significantly improving on a 20-year-old result of Granville and Soundararajan. We also show that the exponent of $\log x$ in this result can be improved to $-1+o(1)$, as long as $f$ does not ``behave like'' the Liouville function $λ$ in a precise sense. As a second application, we show that for a Rademacher random completely multiplicative function $\mathbf{f}$, the probability that $L_{\mathbf{f}}(x)$ is negative is $O(\exp(-x^c))$ for some $c \in (0,1)$, thus establishing a previously conjectured bound. Finally, we obtain a converse theorem for small absolute values $|L_f(x)|$, and construct examples $f$ that show that it is (essentially) best possible.

math.NT↗
arXiv2026

Distribution of mixed character sums and extremal problems for Littlewood polynomials

We prove distributional results for mixed character sums \begin{equation*} \sum_{n\le x }χ(n)e(nθ), \end{equation*} for fixed $θ\in [0,1]$ and random character $χ\pmod q$, as well as for a fixed character $χ$ and randomly sampled $θ\in [0,1].$ We present various applications of our results. For example, we construct Littlewood polynomials with large Mahler measure and $L_1$ norm, thus establishing new records in the Mahler and Newman problems. We also show that $L_{2k}$ norms of well-known Turyn polynomials are asymptotically minimized at the shift $α=1/4,$ proving a conjecture of Günther and Schmidt. An important ingredient in our work is a general way of dealing with "log-integrability" problems.

math.NT↗
arXiv2026

On Artin's conjecture on average and short character sums

Let $N_a(x)$ denote the number of primes up to $x$ for which the integer $a$ is a primitive root. We show that $N_a(x)$ satisfies the asymptotic predicted by Artin's conjecture for almost all $1\le a\le \exp((\log \log x)^2)$. This improves on a result of Stephens (1969). A key ingredient in the proof is a new short character sum estimate over the integers, improving on the range of a result of Garaev (2006).

math.NT↗
arXiv2024

Sign changes of short character sums and real zeros of Fekete polynomials

We discuss a general approach producing quantitative bounds on the number of sign changes of the weighted sums $$\sum_{n\le x}f(n)w_n$$ where $f:\mathbb{N}\to \mathbb{R}$ is a family of multiplicative functions and $w_n\in\mathbb{R}$ are certain weights.As a consequence, we show that for a typical fundamental discriminant $D,$ the partial sums of the real character $χ_D$ change sign $\gg (\log\log D)/\log\log\log \log D$ times on very short initial interval (which goes beyond the range in Vinogradov's conjecture). We also prove that the number of real zeros (localized away from $1$) of the Fekete polynomial associated to a typical fundamental discriminant $D$ is $\gg \frac{\log\log D}{\log\log\log\log D}.$ This comes close to establishing a conjecture of Baker and Montgomery which predicts $\asymp \log \log D$ real zeros. Finally, the same approach shows that almost surely for large $x\ge 1$, the partial sums $\sum_{n\le y}f(n)$ of a (Rademacher) random multiplicative function exhibit $\gg \log \log x/\log \log\log\log x$ sign changes on the interval $[1,x].$ These results rely crucially on uniform quantitative estimates for the joint distribution of $-\frac{L'}{L}(s, χ_D)$ at several points $s$ in the vicinity of the central point $s=1/2$, as well as concentration results for $\log L(s,χ_D)$ in the same range, which we establish. In the second part of the paper, we obtain, for large families of discriminants, ``non-trivial" upper bounds on the number of real zeros of Fekete polynomials, breaking the square root bound. Finally, we construct families of discriminants with associated Fekete polynomials having no zeros away from $1.$

math.NT↗
arXiv2024

Zeros of $L$-functions and large partial sums of Dirichlet coefficients

Let $L(s,π)=\sum_{n=1}^{\infty}λ_π(n)n^{-s}$ be an $L$-function that satisfies a weak form of the generalized Ramanujan conjecture. We prove that large partial sums of $λ_π(n)$ strongly repel the low-lying zeros of $L(s,π)$ away from the critical line. Our results extend and quantitatively improve preceding work of Granville and Soundararajan.

math.NT↗
arXiv2023

Beyond the Erdős discrepancy problem in function fields

We characterize the limiting behavior of partial sums of multiplicative functions $f:\mathbb{F}_q[t]\to S^1$. In contrast to the number field setting, the characterization depends crucially on whether the notion of discrepancy is defined using long intervals, short intervals, or lexicographic intervals. Concerning the notion of short interval discrepancy, we show that a completely multiplicative $f:\mathbb{F}_q[t]\to\{-1,+1\}$ with $q$ odd has bounded short interval sums if and only if $f$ coincides with a "modified" Dirichlet character to a prime power modulus. This confirms the function field version of a conjecture over $\mathbb{Z}$ that such modified characters are extremal with respect to the growth rate of partial sums. Regarding the lexicographic discrepancy, we prove that the discrepancy of a completely multiplicative sequence is always infinite if we define it using a natural lexicographic ordering of $\mathbb{F}_{q}[t]$. This answers a question of Liu and Wooley. Concerning the long sum discrepancy, it was observed by the Polymath 5 collaboration that the Erdős discrepancy problem admits infinitely many completely multiplicative counterexamples on $\mathbb{F}_q[t]$. Nevertheless, we are able to classify the counterexamples if we restrict to the class of modified Dirichlet characters. In this setting, we determine the precise growth rate of the discrepancy, which is still unknown for the analogous problem over the integers.

math.NT↗
arXiv2023

Multiplicative functions in short arithmetic progressions

We study for bounded multiplicative functions $f$ sums of the form \begin{align*} \sum_{\substack{n\leq x \atop n\equiv a\pmod q}}f(n), \end{align*} establishing that their variance over residue classes $a \pmod q$ is small as soon as $q=o(x)$, for almost all moduli $q$, with a nearly power-saving exceptional set of $q$. This improves and generalizes previous results of Hooley on Barban-Davenport-Halberstam-type theorems for such $f$, and moreover our exceptional set is essentially optimal unless one is able to make progress on certain well-known conjectures. We are nevertheless able to prove stronger bounds for the number of the exceptional moduli $q$ in the cases where $q$ is restricted to be either smooth or prime, and conditionally on GRH we show that our variance estimate is valid for every $q$. These results are special cases of a "hybrid result" that works for sums of $f$ over almost all short intervals and arithmetic progressions simultaneously, thus generalizing the Matomäki-Radziwill theorem on multiplicative functions in short intervals. We also consider the maximal deviation of $f$ over all residue classes $a\pmod q$ for $q\leq x^{1/2-\varepsilon}$, and show that it is small for "smooth-supported" $f$, again apart from a nearly power-saving set of exceptional $q$, thus providing a smaller exceptional set than what follows from Bombieri-Vinogradov-type theorems. As an application of our methods, we consider Linnik-type problems for products of exactly three primes, and in particular prove results relating to a ternary version of a conjecture of Erdős on representing every element of the multiplicative group $\mathbb{Z}_p^{\times}$ as the product of two primes less than $p$.

math.NT↗
arXiv2023

$L_q$ norms and Mahler measure of Fekete polynomials

We show that the distribution of the values of Fekete polynomials $F_p$ on the unit circle is governed, as $p\to\infty$, by an explicit limiting (non-Gaussian) random point process.This allows us to prove that the Mahler measure of $F_p$ satisfies $$M_0(F_p)\sim k_0\sqrt{p},$$ as $p\to\infty$ where $k_0=0.74083\dots,$ thus solving an old open problem. Further, we obtain an asymptotic formula for all moments $\|F_p\|_q$ with $0<q<\infty,$ resolving another open problem and improving previous results of Günther and Schmidt (who treated the case $q=2k,$ $k\in\mathbb{N}$).

math.NT↗
arXiv2023

On Elliott's conjecture and applications

Let $f:\mathbb{N}\to \mathbb{D}$ be a multiplicative function. Under the merely necessary assumption that $f$ is non-pretentious (in the sense of Granville and Soundararajan), we show that for any pair of distinct integer shifts $h_1,h_2$ the two-point correlation $$\frac{1}{x}\sum_{n\leq x}{f(n+h_1)\overline{f}(n+h_2)}$$ tends to $0$ along a set of $x\in\mathbb{N}$ of full upper logarithmic density. We also show that the same result holds for the $k$-point correlations $$\frac{1}{x}\sum_{n\leq x}{f(n+h_1)\cdots f(n+h_k)}$$ if $k$ is odd and $f$ is a real-valued non-pretentious function. Previously, the vanishing of correlations was known only under stronger non-pretentiousness hypotheses on $f$ by the works of Tao, and Tao and the third author. We derive several applications, including: (i) A classification of $\pm 1$-valued completely multiplicative functions that omit a length four sign pattern, solving a 1974 conjecture of R.H. Hudson. (ii) A proof that a class of "Liouville-like" functions satisfies the unweighted Elliott conjecture of all orders, solving a problem of de la Rue. (iii) Constructing examples of multiplicative $f:\mathbb{N}\to \{-1,0,1\}$ with a given (unique) Furstenberg system, answering a question of Lemańczyk. (iv) A density version of the Erdős discrepancy theorem of Tao.

math.NT↗
arXiv2022

Correlations of multiplicative functions in function fields

We develop an approach to study character sums, weighted by a multiplicative function $f:\mathbb{F}_q[t]\to S^1$, of the form \begin{equation} \sum_{G\in \mathcal{M}_N}f(G)χ(G)ξ(G), \end{equation} where $χ$ is a Dirichlet character and $ξ$ is a short interval character over $\mathbb{F}_q[t].$ We then deduce versions of the Matomäki-Radziwill theorem and Tao's two-point logarithmic Elliott conjecture over function fields $\mathbb{F}_q[t]$, where $q$ is fixed. The former of these improves on work of Gorodetsky, and the latter extends the work of Sawin-Shusterman on correlations of the Möbius function for various values of $q$. Compared with the integer setting, we encounter a different phenomenon, specifically a low characteristic issue in the case that $q$ is a power of $2$. As an application of our results, we give a short proof of the function field version of a conjecture of Kátai on classifying multiplicative functions with small increments, with the classification obtained and the proof being different from the integer case. In a companion paper, we use these results to characterize the limiting behavior of partial sums of multiplicative functions in function fields and in particular to solve a "corrected" form of the Erdős discrepancy problem over $\mathbb{F}_q[t]$.

math.NT↗
arXiv2022

How negative can $\sum_{n\le x}\frac{f(n)}{n}$ be?

Turán observed that logarithmic partial sums $\sum_{n\le x}\frac{f(n)}{n}$ of completely multiplicative functions (in the particular case of the Liouville function $f(n)=λ(n)$) tend to be positive. We develop a general approach to prove two results aiming to explain this phenomena. Firstly, we show that for every $\varepsilon>0$ there exists some $x_0\ge 1,$ such that for any completely multiplicative function $f$ satisfying $-1\le f(n)\le 1$, we have $$\sum_{n\le x}\frac{f(n)}{n}\ge -\frac{1}{(\log\log{x})^{1-\varepsilon}}, \quad x\ge x_0.$$ This improves a previous bound due to Granville and Soundararajan. Secondly, we show that if $f$ is a typical (random) completely multiplicative function $f:\mathbb{N}\to \{-1,1\}$, the probability that $\sum_{n\le x}\frac{f(n)}{n}$ is negative for a given large $x,$ is $O(\exp(-\exp(\frac{\log x\cdot \log\log\log x}{C\log \log x}))).$ This improves on recent work of Angelo and Xu.

math.NT↗
arXiv2022

On the random Chowla conjecture

We show that for a Steinhaus random multiplicative function $f:\mathbb{N}\to\mathbb{D}$ and any polynomial $P(x)\in\mathbb{Z}[x]$ of $\text{deg}\ P\ge 2$ which is not of the form $w(x+c)^{d}$ for some $w\in \mathbb{Z}$, $c\in \mathbb{Q}$, we have \[\frac{1}{\sqrt{x}}\sum_{n\le x} f(P(n)) \xrightarrow{d} \mathcal{CN}(0,1),\] where $\mathcal{CN}(0,1)$ is the standard complex Gaussian distribution with mean $0$ and variance $1.$ This confirms a conjecture of Najnudel in a strong form. We further show that there almost surely exist arbitrary large values of $x\ge 1,$ such that $$|\sum_{n\le x} f(P(n))| \gg_{\text{deg}\ P} \sqrt{x} (\log \log x)^{1/2},$$ for any polynomial $P(x)\in\mathbb{Z}[x]$ with $\text{deg}\ P\ge 2,$ which is not a product of linear factors (over $\mathbb{Q}$). This matches the bound predicted by the law of the iterated logarithm. Both of these results are in contrast with the well-known case of linear phase $P(n)=n,$ where the partial sums are known to behave in a non-Gaussian fashion and the corresponding sharp fluctuations are speculated to be $O(\sqrt{x}(\log \log x)^{\frac{1}{4}+\varepsilon})$ for any $\varepsilon>0$.

math.NT↗
arXiv2021

On the behavior of nodal lines near the boundary for Laplace eigenfunctions on the square

We are interested in the effect of Dirichlet boundary conditions on the nodal length of Laplace eigenfunctions. We study random Gaussian Laplace eigenfunctions on the two dimensional square and find a two terms asymptotic expansion for the expectation of the nodal length in any square of side larger than the Planck scale, along a denisty one sequence of energy levels. The proof relies on a new study of lattice points in small arcs, and shows that the said expectation is independent of the position of the square, giving the same asymptotic expansion both near and far from the boundaries.

math.PR↗