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Maksym Radziwiłł

Publications and source records attributed to Maksym Radziwiłł.

50 records · Page 3Linked to original sources
arXiv2018

Level of distribution of unbalanced convolutions

We show that if an essentially arbitrary sequence supported on an interval containing $x$ integers, is convolved with a tiny Siegel-Walfisz-type sequence supported on an interval containing $\exp((\log x)^{\varepsilon})$ integers then the resulting multiplicative convolution has (in a weak sense) level of distribution $x^{1/2 + 1/66 - \varepsilon}$ as $x$ goes to infinity. This dispersion estimate has a number of consequences for: the distribution of the $k$th divisor function to moduli $x^{1/2 + 1/66 - \varepsilon}$ for any integer $k \geq 1$, the distribution of products of exactly two primes in arithmetic progressions to large moduli, the distribution of sieve weights of level $x^{1/2 + 1/66 - \varepsilon}$ to moduli as large as $x^{1 - \varepsilon}$ and for the Brun-Titchmarsh theorem for almost all moduli $q$ of size $x^{1 - \varepsilon}$, lowering the long-standing constant $4$ in that range. Our result improves and is inspired by earlier work of Green (and subsequent work of Granville-Shao) which is concerned with the distribution of $1$-bounded multiplicative functions in arithmetic progressions to large moduli. As in these previous works the main technical ingredient are the recent estimates of Bettin-Chandee for trilinear forms in Kloosterman fractions and the estimates of Duke-Friedlander-Iwaniec for bilinear forms in Kloosterman fractions.

math.NT↗
arXiv2018

Triple correlation and long gaps in the spectrum of flat tori

We evaluate the triple correlation of eigenvalues of the Laplacian on generic flat tori in an averaged sense. As two consequence we show that (a) the limit inferior (resp. limit superior) of the triple correlation is almost surely at most (resp. at least) Poissonian, and that (b) almost all flat tori contain infinitely many gaps in their spectrum that are at least 2.006 times longer than the average gap. The significance of the constant 2.006 lies in the fact that there exist sequences with Poissonian pair correlation and with consecutive spacings bounded uniformly from above by 2, as we also prove in this paper. Thus our result goes beyond what can be deduced solely from the Poissonian behavior of the pair correlation.

math.NT↗
arXiv2018

Limiting distribution of eigenvalues in the large sieve matrix

The large sieve inequality is equivalent to the bound $λ_1 \leqslant N + Q^2-1$ for the largest eigenvalue $λ_1$ of the $N$ by $N$ matrix $A^{\star} A$, naturally associated to the positive definite quadratic form arising in the inequality. For arithmetic applications the most interesting range is $N \asymp Q^2$. Based on his numerical data Ramaré conjectured that when $N \sim αQ^2$ as $Q \rightarrow \infty$ for some finite positive constant $α$, the limiting distribution of the eigenvalues of $A^{\star} A$, scaled by $1/N$, exists and is non-degenerate. In this paper we prove this conjecture by establishing the convergence of all moments of the eigenvalues of $A^{\star} A$ as $Q\rightarrow\infty$. Previously only the second moment was known, due to Ramaré. Furthermore, we obtain an explicit description of the moments of the limiting distribution, and establish that they vary continuously with $α$. Some of the main ingredients in our proof include the large-sieve inequality and results on $n$-correlations of Farey fractions.

math.NT↗
arXiv2018

Maximum of the Riemann zeta function on a short interval of the critical line

We prove the leading order of a conjecture by Fyodorov, Hiary and Keating, about the maximum of the Riemann zeta function on random intervals along the critical line. More precisely, as $T \rightarrow \infty$ for a set of $t \in [T, 2T]$ of measure $(1 - o(1)) T$, we have $$ \max_{|t-u|\leq 1}\log\left|ζ\left(\tfrac{1}{2}+i u\right)\right|=(1 + o(1))\log\log T . $$

math.PR↗
arXiv2017

Multiplicative functions in short intervals

We introduce a general result relating "short averages" of a multiplicative function to "long averages" which are well understood. This result has several consequences. First, for the Möbius function we show that there are cancellations in the sum of $μ(n)$ in almost all intervals of the form $[x, x + ψ(x)]$ with $ψ(x) \rightarrow \infty$ arbitrarily slowly. This goes beyond what was previously known conditionally on the Density Hypothesis or the stronger Riemann Hypothesis. Second, we settle the long-standing conjecture on the existence of $x^ε$-smooth numbers in intervals of the form $[x, x + c(\varepsilon) \sqrt{x}]$, recovering unconditionally a conditional (on the Riemann Hypothesis) result of Soundararajan. Third, we show that the mean-value of $λ(n)λ(n+1)$, with $λ(n)$ Liouville's function, is non-trivially bounded in absolute value by $1 - δ$ for some $δ> 0$. This settles an old folklore conjecture and constitutes progress towards Chowla's conjecture. Fourth, we show that a (general) real-valued multiplicative function $f$ has a positive proportion of sign changes if and only if $f$ is negative on at least one integer and non-zero on a positive proportion of the integers. This improves on many previous works, and is new already in the case of the Möbius function. We also obtain some additional results on smooth numbers in almost all intervals, and sign changes of multiplicative functions in all intervals of square-root length.

math.NT↗
arXiv2016

Small gaps in the spectrum of the rectangular billiard

We study the size of the minimal gap between the first N eigenvalues of the Laplacian on a rectangular billiard having irrational squared aspect ratio $α$, in comparison to the corresponding quantity for a Poissonian sequence. If $α$ is a quadratic irrationality of certain type, such as the square root of a rational number, we show that the minimal gap is roughly of size 1/N, which is essentially consistent with Poisson statistics. We also give related results for a set of $α$'s of full measure. However, on a fine scale we show that Poisson statistics is violated for all $α$. The proofs use a variety of ideas of an arithmetical nature, involving Diophantine approximation, the theory of continued fractions, and results in analytic number theory.

math.AP↗
arXiv2016

A quadratic divisor problem and moments of the Riemann zeta-function

We estimate asymptotically the fourth moment of the Riemann zeta-function twisted by a Dirichlet polynomial of length $T^{\frac14 - \varepsilon}$. Our work relies crucially on Watt's theorem on averages of Kloosterman fractions. In the context of the twisted fourth moment, Watt's result is an optimal replacement for Selberg's eigenvalue conjecture. Our work extends the previous result of Hughes and Young, where Dirichlet polynomials of length $T^{\frac{1}{11}-\varepsilon}$ were considered. Our result has several applications, among others to the proportion of critical zeros of the Riemann zeta-function, zero spacing and lower bounds for moments. Along the way we obtain an asymptotic formula for a quadratic divisor problem, where the condition $a m_1 m_2 - b n_1 n_2 = h$ is summed with smooth averaging on the variables $m_1, m_2, n_1, n_2, h$ and arbitrary weights in the average on $a,b$. Using Watt's work allows us to exploit all averages simultaneously. It turns out that averaging over $m_1, m_2, n_1, n_2, h$ right away in the quadratic divisor problem simplifies considerably the combinatorics of the main terms in the twisted fourth moment.

math.NT↗
arXiv2016

Quantum Unique Ergodicity for half-integral weight forms

We investigate the analogue of the Quantum Unique Ergodicity (QUE) conjecture for half-integral weight automorphic forms. Assuming the Generalized Riemann Hypothesis (GRH) we establish QUE for both half-integral weight holomorphic Hecke cusp forms for $Γ_0(4)$ lying in Kohnen's plus subspace and for half-integral weight Hecke Maaß cusp forms for $Γ_0(4)$ lying in Kohnen's plus subspace. By combining the former result along with an argument of Rudnick, it follows that under GRH the zeros of these holomorphic Hecke cusp equidistribute with respect to hyperbolic measure on $Γ_0(4)\backslash \mathbb H$ as the weight tends to infinity.

math.NT↗
arXiv2015

Sign patterns of the Liouville and Möbius functions

Let $λ$ and $μ$ denote the Liouville and Möbius functions respectively. Hildebrand showed that all eight possible sign patterns for $(λ(n), λ(n+1), λ(n+2))$ occur infinitely often. By using the recent result of the first two authors on mean values of multiplicative functions in short intervals, we strengthen Hildebrand's result by proving that each of these eight sign patterns occur with positive lower natural density. We also obtain an analogous result for the nine possible sign patterns for $(μ(n), μ(n+1))$. A new feature in the latter argument is the need to demonstrate that a certain random graph is almost surely connected.

math.NT↗
arXiv2015

Small scale distribution of zeros and mass of modular forms

We study the behavior of zeros and mass of holomorphic Hecke cusp forms on $SL_2(\mathbb Z) \backslash \mathbb H$ at small scales. In particular, we examine the distribution of the zeros within hyperbolic balls whose radii shrink sufficiently slowly as $k \rightarrow \infty$. We show that the zeros equidistribute within such balls as $k \rightarrow \infty$ as long as the radii shrink at a rate at most a small power of $1/\log k$. This relies on a new, effective, proof of Rudnick's theorem on equidistribution of the zeros and on an effective version of Quantum Unique Ergodicity for holomorphic forms, which we obtain in this paper. We also examine the distribution of the zeros near the cusp of $SL_2(\mathbb Z) \backslash \mathbb H$. Ghosh and Sarnak conjectured that almost all the zeros here lie on two vertical geodesics. We show that for almost all forms a positive proportion of zeros high in the cusp do lie on these geodesics. For all forms, we assume the Generalized Lindelöf Hypothesis and establish a lower bound on the number of zeros that lie on these geodesics, which is significantly stronger than the previous unconditional results.

math.NT↗
arXiv2015

A note on Helson's conjecture on moments of random multiplicative functions

We give lower bounds for the small moments of the sum of a random multiplicative function, which improve on some results of Bondarenko and Seip and constitute further progress towards (dis)proving a conjecture of Helson. We also prove asymptotics for the even integer moments. The latter have also been obtained very recently and independently by Heap and Lindqvist. Our proofs involve general lower bound techniques for random multiplicative functions, mean value results for multiplicative functions in several variables, and some calculations with Birkhoff polytopes.

math.NT↗
arXiv2015

A note on the Liouville function in short intervals

In this note we give a short and self-contained proof that, for any $δ> 0$, $\sum_{x \leq n \leq x+x^δ} λ(n) = o(x^δ)$ for almost all $x \in [X, 2X]$. We also sketch a proof of a generalization of such a result to general real-valued multiplicative functions. Both results are special cases of results in our more involved and lengthy recent pre-print.

math.NT↗
arXiv2013

Simple zeros of primitive Dirichlet $L$-functions and the asymptotic large sieve

Assuming the Generalized Riemann Hypothesis (GRH), we show using the asymptotic large sieve that 91% of the zeros of primitive Dirichlet $L$-functions are simple. This improves on earlier work of Özlük which gives a proportion of at most 86%. We further compute an $q$-analogue of the Pair Correlation Function $F(α)$ averaged over all primitive Dirichlet $L$-functions in the range $|α| < 2$ . Previously such a result was available only when the average included all the characters $χ$.

math.NT↗