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Maksym Radziwiłł

Publications and source records attributed to Maksym Radziwiłł.

At least 37 records · Page 2Linked to original sources
arXiv2022

Higher uniformity of bounded multiplicative functions in short intervals on average

Let $λ$ denote the Liouville function. We show that, as $X \rightarrow \infty$, $$\int_{X}^{2X} \sup_{\substack{P(Y)\in \mathbb{R}[Y]\\ deg(P)\leq k}} \Big | \sum_{x \leq n \leq x + H} λ(n) e(-P(n)) \Big |\ dx = o ( X H)$$ for all fixed $k$ and $X^θ \leq H \leq X$ with $0 < θ< 1$ fixed but arbitrarily small. Previously this was only established for $k \leq 1$. We obtain this result as a special case of the corresponding statement for (non-pretentious) $1$-bounded multiplicative functions that we prove. In fact, we are able to replace the polynomial phases $e(-P(n))$ by degree $k$ nilsequences $\overline{F}(g(n) Γ)$. By the inverse theory for the Gowers norms this implies the higher order asymptotic uniformity result $$\int_{X}^{2X} \| λ\|_{U^{k+1}([x,x+H])}\ dx = o ( X )$$ in the same range of $H$. We present applications of this result to patterns of various types in the Liouville sequence. Firstly, we show that the number of sign patterns of the Liouville function is superpolynomial, making progress on a conjecture of Sarnak about the Liouville sequence having positive entropy. Secondly, we obtain cancellation in averages of $λ$ over short polynomial progressions $(n+P_1(m),\ldots, n+P_k(m))$, which in the case of linear polynomials yields a new averaged version of Chowla's conjecture. We are in fact able to prove our results on polynomial phases in the wider range $H\geq \exp((\log X)^{5/8+\varepsilon})$, thus strengthening also previous work on the Fourier uniformity of the Liouville function.

math.NT↗
arXiv2022

An averaged form of Chowla's conjecture

Let $λ$ denote the Liouville function. A well known conjecture of Chowla asserts that for any distinct natural numbers $h_1,\dots,h_k$, one has $\sum_{1 \leq n \leq X} λ(n+h_1) \dotsm λ(n+h_k) = o(X)$ as $X \to \infty$. This conjecture remains unproven for any $h_1,\dots,h_k$ with $k \geq 2$. In this paper, using the recent results of the first two authors on mean values of multiplicative functions in short intervals, combined with an argument of Katai and Bourgain-Sarnak-Ziegler, we establish an averaged version of this conjecture, namely $$\sum_{h_1,\dots,h_k \leq H} \left|\sum_{1 \leq n \leq X} λ(n+h_1) \dotsm λ(n+h_k)\right| = o(H^kX)$$ as $X \to \infty$ whenever $H = H(X) \leq X$ goes to infinity as $X \to \infty$, and $k$ is fixed. Related to this, we give the exponential sum estimate $$ \int_0^X \left|\sum_{x \leq n \leq x+H} λ(n) e(αn)\right| dx = o( HX )$$ as $X \to \infty$ uniformly for all $α\in \mathbb{R}$, with $H$ as before. Our arguments in fact give quantitative bounds on the decay rate (roughly on the order of $\frac{\log\log H}{\log H}$), and extend to more general bounded multiplicative functions than the Liouville function, yielding an averaged form of a (corrected) conjecture of Elliott.

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arXiv2021

Small gaps in the spectrum of tori: asymptotic formulae

We establish an asymptotic formula, uniformly down to the Planck scale, for the number of small gaps between the first N eigenvalues of the Laplacian on almost all flat tori and also on almost all rectangular flat tori.

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arXiv2021

Singmaster's conjecture in the interior of Pascal's triangle

Singmaster's conjecture asserts that every natural number greater than one occurs at most a bounded number of times in Pascal's triangle; that is, for any natural number $t \geq 2$, the number of solutions to the equation $\binom{n}{m} = t$ for natural numbers $1 \leq m < n$ is bounded. In this paper we establish this result in the interior region $\exp(\log^{2/3+\varepsilon} n) \leq m \leq n-\exp(\log^{2/3 + \varepsilon} n)$ for any fixed $\varepsilon > 0$. Indeed, when $t$ is sufficiently large depending on $\varepsilon$, we show that there are at most four solutions (or at most two in either half of Pascal's triangle) in this region. We also establish analogous results for the equation $(n)_m = t$, where $(n)_m := n(n-1)\ldots(n-m+1)$ denotes the falling factorial.

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arXiv2021

Optimal Small Scale Equidistribution of Lattice Points on the Sphere, Heegner Points, and Closed Geodesics

We asymptotically estimate the variance of the number of lattice points in a thin, randomly rotated annulus lying on the surface of the sphere. This partially resolves a conjecture of Bourgain, Rudnick, and Sarnak. We also obtain estimates that are valid for all balls and annuli that are not too small. Our results have several consequences: for a conjecture of Linnik on sums of two squares and a "microsquare", a conjecture of Bourgain and Rudnick on the number of lattice points lying in small balls on the surface of the sphere, the covering radius of the sphere, and the distribution of lattice points in almost all thin regions lying on the surface of the sphere. Finally, we show that for a density $1$ subsequence of squarefree integers, the variance exhibits a different asymptotic behaviour for balls of volume $(\log n)^{-δ}$ with $0 < δ< \tfrac {1}{16}$. We also obtain analogous results for Heegner points and closed geodesics. Interestingly, we are able to prove some slightly stronger results for closed geodesics than for Heegner points or lattice points on the surface of the sphere. A crucial observation that underpins our proof is the different behaviour of weights functions for annuli and for balls.

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arXiv2021

Moments of the Riemann zeta function on short intervals of the critical line

We show that as $T\to \infty$, for all $t\in [T,2T]$ outside of a set of measure $\mathrm{o}(T)$, $$ \int_{-(\log T)^θ}^{(\log T)^θ} |ζ(\tfrac 12 + \mathrm{i} t + \mathrm{i} h)|^β \mathrm{d} h = (\log T)^{f_θ(β) + \mathrm{o}(1)}, $$ for some explicit exponent $f_θ(β)$, where $θ> -1$ and $β> 0$. This proves an extended version of a conjecture of Fyodorov and Keating (2014). In particular, it shows that, for all $θ> -1$, the moments exhibit a phase transition at a critical exponent $β_c(θ)$, below which $f_θ(β)$ is quadratic and above which $f_θ(β)$ is linear. The form of the exponent $f_θ$ also differs between mesoscopic intervals ($-1<θ<0$) and macroscopic intervals ($θ>0$), a phenomenon that stems from an approximate tree structure for the correlations of zeta. We also prove that, for all $t\in [T,2T]$ outside a set of measure $\mathrm{o}(T)$, $$ \max_{|h| \leq (\log T)^θ} |ζ(\tfrac{1}{2} + \mathrm{i} t + \mathrm{i} h)| = (\log T)^{m(θ) + \mathrm{o}(1)}, $$ for some explicit $m(θ)$. This generalizes earlier results of Najnudel (2018) and Arguin et al. (2019) for $θ= 0$. The proofs are unconditional, except for the upper bounds when $θ> 3$, where the Riemann hypothesis is assumed.

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arXiv2021

Expansion, divisibility and parity

Let $\mathbf{P} \subset [H_0,H]$ be a set of primes, where $\log H_0 \geq (\log H)^{2/3 + ε}$. Let $\mathscr{L} = \sum_{p \in \mathbf{P}} 1/p$. Let $N$ be such that $\log H \leq (\log N)^{1/2-ε}$. We show there exists a subset $\mathscr{X} \subset (N, 2N]$ of density close to $1$ such that all the eigenvalues of the linear operator $$(A_{|\mathscr{X}} f)(n) = \sum_{\substack{p \in \mathbf{P} : p | n \\ n, n \pm p \in \mathscr{X}}} f(n \pm p) \; - \sum_{\substack{p \in\mathbf{P} \\ n, n \pm p \in \mathscr{X}}} \frac{f(n \pm p)}{p}$$ are $O(\sqrt{\mathscr{L}})$. This bound is optimal up to a constant factor. In other words, we prove that a graph describing divisibility by primes is a strong local expander almost everywhere, and indeed within a constant factor of being "locally Ramanujan" (a.e.). Specializing to $f(n) = λ(n)$ with $λ(n)$ the Liouville function, and using an estimate by Matomäki, Radziwiłł and Tao on the average of $λ(n)$ in short intervals, we derive that \[\frac{1}{\log x} \sum_{n\leq x} \frac{λ(n) λ(n+1)}{n} = O\Big(\frac{1}{\sqrt{\log \log x}}\Big),\] improving on a result of Tao's. We also prove that $\sum_{N<n\leq 2 N} λ(n) λ(n+1)=o(N)$ at almost all scales with a similar error term, improving on a result by Tao and Teräväinen. (Tao and Tao-Teräväinen followed a different approach, based on entropy, not expansion; significantly, we can take a much larger value of $H$, and thus consider many more primes.) We can also prove sharper results with ease. For instance: let $S_{N,k}$ the set of all $N<n\leq 2N$ such that $Ω(n) = k$. Then, for any fixed value of $k$ with $k = \log \log N + O(\sqrt{\log \log N})$ (that is, any "popular" value of $k$) the average of $λ(n+1)$ over $S_{N,k}$ is $o(1)$ at almost all scales.

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arXiv2020

On the variance of squarefree integers in short intervals and arithmetic progressions

We evaluate asymptotically the variance of the number of squarefree integers up to $x$ in short intervals of length $H < x^{6/11 - \varepsilon}$ and the variance of the number of squarefree integers up to $x$ in arithmetic progressions modulo $q$ with $q > x^{5/11 + \varepsilon}$. On the assumption of respectively the Lindelöf Hypothesis and the Generalized Lindelöf Hypothesis we show that these ranges can be improved to respectively $H < x^{2/3 - \varepsilon}$ and $q > x^{1/3 + \varepsilon}$. Furthermore we show that obtaining a bound sharp up to factors of $H^{\varepsilon}$ in the full range $H < x^{1 - \varepsilon}$ is equivalent to the Riemann Hypothesis. These results improve on a result of Hall (1982) for short intervals, and earlier results of Warlimont, Vaughan, Blomer, Nunes and Le Boudec in the case of arithmetic progressions.

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arXiv2020

The Fyodorov-Hiary-Keating Conjecture. I

By analogy with conjectures for random matrices, Fyodorov-Hiary-Keating and Fyodorov-Keating proposed precise asymptotics for the maximum of the Riemann zeta function in a typical short interval on the critical line. In this paper, we settle the upper bound part of their conjecture in a strong form. More precisely, we show that the measure of those $T \leq t \leq 2T$ for which $$ \max_{|h| \leq 1} |ζ(1/2 + i t + i h)| > e^y \frac{\log T }{(\log\log T)^{3/4}}$$ is bounded by $Cy e^{-2y}$ uniformly in $y \geq 1$. This is expected to be optimal for $y= O(\sqrt{\log\log T})$. This upper bound is sharper than what is known in the context of random matrices, since it gives (uniform) decay rates in $y$. In a subsequent paper we will obtain matching lower bounds.

math.PR↗
arXiv2020

Multiplicative functions in short intervals II

We determine the behavior of multiplicative functions vanishing at a positive proportion of prime numbers in almost all short intervals. Furthermore we quantify "almost all" with uniform power-saving upper bounds, that is, we save a power of the suitably normalized length of the interval regardless of how long or short the interval is. Such power-saving bounds are new even in the special case of the Möbius function. These general results are motivated by several applications. First, we strengthen work of Hooley on sums of two squares by establishing an asymptotic for the number of integers that are sums of two squares in almost all short intervals. Previously only the order of magnitude was known. Secondly, we extend this result to general norm forms of an arbitrary number field $K$ (sums of two squares are norm-forms of $\mathbb{Q}(i)$). Thirdly, Hooley determined the order of magnitude of the sum of $(s_{n + 1} - s_{n})^γ$ with $γ\in (1, 5/3)$ where $s_{1} < s_2 < \ldots$ denote integers representable as sums of two squares. We establish a similar results with $γ\in (1, 3/2)$ and $s_n$ the sequence of integers representable as norm-forms of an arbitrary number field $K$. This is the first such result for a number field of degree greater than two. Assuming the Riemann Hypothesis for all Hecke $L$-functions we also show that $γ\in (1,2)$ is admissible. Fourthly, we improve on a recent result of Heath-Brown about gaps between $x^{\varepsilon}$-smooth numbers. More generally, we obtain results about gaps between multiplicative sequences. Finally our result is useful in other contexts aswell, for instance in our forthcoming work on Fourier uniformity (joint with Terence Tao, Joni Teraväinen and Tamar Ziegler).

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arXiv2020

Prime number theorem for analytic skew products

We establish a prime number theorem for all uniquely ergodic, analytic skew products on the $2$-torus $\mathbb{T}^2$. More precisely, for every irrational $α$ and every $1$-periodic real analytic $g:\mathbb{R}\to\mathbb{R}$ of zero mean, let $T_{α,g} : \mathbb{T}^2 \rightarrow \mathbb{T}^2$ be defined by $(x,y) \mapsto (x+α,y+g(x))$. We prove that if $T_{α, g}$ is uniquely ergodic then, for every $(x,y) \in \mathbb{T}^2$, the sequence $\{T_{α, g}^p(x,y)\}$ is equidistributed on $\mathbb{T}^2$ as $p$ traverses prime numbers. This is the first example of a class of natural, non-algebraic and smooth dynamical systems for which a prime number theorem holds. We also show that such a prime number theorem does not necessarily hold if $g$ is only continuous on $\mathbb{T}^2$.

math.DS↗
arXiv2020

Correlations of the von Mangoldt and higher divisor functions II. Divisor correlations in short ranges

We study the problem of obtaining asymptotic formulas for the sums $\sum_{X < n \leq 2X} d_k(n) d_l(n+h)$ and $\sum_{X < n \leq 2X} Λ(n) d_k(n+h)$, where $Λ$ is the von Mangoldt function, $d_k$ is the $k^{\operatorname{th}}$ divisor function, $X$ is large and $k \geq l \geq 2$ are real numbers. We show that for almost all $h \in [-H, H]$ with $H = (\log X)^{10000 k \log k}$, the expected asymptotic estimate holds. In our previous paper we were able to deal also with the case of $Λ(n) Λ(n + h)$ and we obtained better estimates for the error terms at the price of having to take $H = X^{8/33 + \varepsilon}$.

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arXiv2019

Signs of Fourier coefficients of half-integral weight modular forms

Let $g$ be a Hecke cusp form of half-integral weight, level $4$ and belonging to Kohnen's plus subspace. Let $c(n)$ denote the $n$th Fourier coefficient of $g$, normalized so that $c(n)$ is real for all $n \geq 1$. A theorem of Waldspurger determines the magnitude of $c(n)$ at fundamental discriminants $n$ by establishing that the square of $c(n)$ is proportional to the central value of a certain $L$-function. The signs of the sequence $c(n)$ however remain mysterious. Conditionally on the Generalized Riemann Hypothesis, we show that $c(n) < 0$ and respectively $c(n) > 0$ holds for a positive proportion of fundamental discriminants $n$. Moreover we show that the sequence $\{c(n)\}$ where $n$ ranges over fundamental discriminants changes sign a positive proportion of the time. Unconditionally, it is not known that a positive proportion of these coefficients are non-zero and we prove results about the sign of $c(n)$ which are of the same quality as the best known non-vanishing results. Finally we discuss extensions of our result to general half-integral weight forms $g$ of level $4 N$ with $N$ odd, square-free.

math.NT↗
arXiv2019

Correlations of the von Mangoldt and higher divisor functions I. Long shift ranges

We show that the expected asymptotic for the sums $\sum_{X < n \leq 2X} Λ(n) Λ(n+h)$, $\sum_{X < n \leq 2X} d_k(n) d_l(n+h)$, and $\sum_{X < n \leq 2X} Λ(n) d_k(n+h)$ hold for almost all $h \in [-H,H]$, provided that $X^{8/33+\varepsilon} \leq H \leq X^{1-\varepsilon}$, with an error term saving on average an arbitrary power of the logarithm over the trivial bound. Previous work of Mikawa, Perelli-Pintz and Baier-Browning-Marasingha-Zhao covered the range $H \geq X^{1/3+\varepsilon}$. We also obtain an analogous result for $\sum_n Λ(n) Λ(N-n)$. Our proof uses the circle method and some oscillatory integral estimates (following a paper of Zhan) to reduce matters to establishing some mean-value estimates for certain Dirichlet polynomials associated to "Type $d_3$" and "Type $d_4$" sums (as well as some other sums that are easier to treat). After applying Hölder's inequality to the Type $d_3$ sum, one is left with two expressions, one of which we can control using a short interval mean value theorem of Jutila, and the other we can control using exponential sum estimates of Robert and Sargos. The Type $d_4$ sum is treated similarly using the classical $L^2$ mean value theorem and the classical van der Corput exponential sum estimates.

math.NT↗
arXiv2018

A note on the dimension of the largest simple Hecke submodule

For $k\ge 2$ even, let $d_{k,N}$ denote the dimension of the largest simple Hecke submodule of $S_{k}(Γ_0(N); \mathbb{Q})^\text{new}$. We show, using a simple analytic method, that $d_{k,N} \gg_k \log\log N / \log(2p)$ with $p$ the smallest prime co-prime to $N$. Previously, bounds of this quality were only known for $N$ in certain subsets of the primes. We also establish similar (and sometimes stronger) results concerning $S_{k}(Γ_0(N), χ)$, with $k \geq 2$ an integer and $χ$ an arbitrary nebentypus.

math.NT↗
arXiv2018

Another application of Linnik's dispersion method

Let $α_m$ and $β_n$ be two sequences of real numbers supported on $[M, 2M]$ and $[N, 2N]$ with $M = X^{1/2 - δ}$ and $N = X^{1/2 + δ}$. We show that there exists a $δ_0 > 0$ such that the multiplicative convolution of $α_m$ and $β_n$ has exponent of distribution $\frac{1}{2} + δ-\varepsilon$ (in a weak sense) as long as $0 \leq δ< δ_0$, the sequence $β_n$ is Siegel-Walfisz and both sequences $α_m$ and $β_n$ are bounded above by divisor functions. Our result is thus a general dispersion estimate for "narrow" type-II sums. The proof relies crucially on Linnik's dispersion method and recent bounds for trilinear forms in Kloosterman fractions due to Bettin-Chandee. We highlight an application related to the Titchmarsh divisor problem.

math.NT↗
arXiv2018

Fourier uniformity of bounded multiplicative functions in short intervals on average

Let $λ$ denote the Liouville function. We show that as $X \rightarrow \infty$, $$ \int_{X}^{2X} \sup_α \left | \sum_{x < n \leq x + H} λ(n) e(-αn) \right | dx = o ( X H) $$ for all $H \geq X^θ$ with $θ> 0$ fixed but arbitrarily small. Previously, this was only known for $θ> 5/8$. For smaller values of $θ$ this is the first `non-trivial' case of local Fourier uniformity on average at this scale. We also obtain the analogous statement for (non-pretentious) $1$-bounded multiplicative functions. We illustrate the strength of the result by obtaining cancellations in the sum of $λ(n) Λ(n + h) Λ(n + 2h)$ over the ranges $h < X^θ$ and $n < X$, and where $Λ$ is the von Mangoldt function.

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