Monograph records
https://feeds.library.caltech.edu/people/Radziwiłł-M/monograph.rss
A Caltech Library Repository Feedhttp://www.rssboard.org/rss-specificationpython-feedgenenTue, 16 Apr 2024 14:10:05 +0000Maximum of the Riemann zeta function on a short interval of the critical line
https://resolver.caltech.edu/CaltechAUTHORS:20180612-140618985
Authors: {'items': [{'id': 'Arguin-L-P', 'name': {'family': 'Arguin', 'given': 'Louis-Pierre'}}, {'id': 'Belius-D', 'name': {'family': 'Belius', 'given': 'David'}}, {'id': 'Bourgade-P', 'name': {'family': 'Bourgade', 'given': 'Paul'}}, {'id': 'Radziwiłł-M', 'name': {'family': 'Radziwiłł', 'given': 'Maksym'}}, {'id': 'Soundararajan-K', 'name': {'family': 'Soundararajan', 'given': 'Kannan'}}]}
Year: 2018
DOI: 10.48550/arXiv.1612.08575
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→∞ for a set of t∈[T,2T] of measure (1−o(1))T, we have max|t−u|≤1log∣∣ζ(12+iu)∣∣=(1+o(1))loglogT.https://authors.library.caltech.edu/records/16cwf-trs17Correlations of the von Mangoldt and higher divisor functions I. Long shift ranges
https://resolver.caltech.edu/CaltechAUTHORS:20180612-134458335
Authors: {'items': [{'id': 'Matomäki-K', 'name': {'family': 'Matomäki', 'given': 'Kaisa'}}, {'id': 'Radziwiłł-M', 'name': {'family': 'Radziwiłł', 'given': 'Maksym'}}, {'id': 'Tao-Terence', 'name': {'family': 'Tao', 'given': 'Terence'}}]}
Year: 2018
DOI: 10.48550/arXiv.1707.01315
We show that the expected asymptotic for the sums ∑_(Xhttps://authors.library.caltech.edu/records/as59c-at949An effective universality theorem for the Riemann zeta-function
https://resolver.caltech.edu/CaltechAUTHORS:20180612-153643797
Authors: {'items': [{'id': 'Lamzouri-Y', 'name': {'family': 'Lamzouri', 'given': 'Youness'}}, {'id': 'Lester-S', 'name': {'family': 'Lester', 'given': 'Stephen'}, 'orcid': '0000-0003-1977-9205'}, {'id': 'Radziwiłł-M', 'name': {'family': 'Radziwiłł', 'given': 'Maksym'}}]}
Year: 2018
DOI: 10.48550/arXiv.1611.10325
Let 00, once T is large enough. This was refined by Bagchi who showed that the measure of such t∈[0,T] is (c(ε)+o(1))T, for all but at most countably many ε>0. Using a completely different approach, we obtain the first effective version of Voronin's Theorem, by showing that in the rate of convergence one can save a small power of the logarithm of T. Our method is flexible, and can be generalized to other L-functions in the t-aspect, as well as to families of L-functions in the conductor aspect.https://authors.library.caltech.edu/records/146qt-3sd67A note on the Liouville function in short intervals
https://resolver.caltech.edu/CaltechAUTHORS:20180613-084923189
Authors: {'items': [{'id': 'Matomäki-K', 'name': {'family': 'Matomäki', 'given': 'Kaisa'}}, {'id': 'Radziwiłł-M', 'name': {'family': 'Radziwiłł', 'given': 'Maksym'}}]}
Year: 2018
DOI: 10.48550/arXiv.1502.02374
In this note we give a short and self-contained proof that, for any δ > 0, ∑_(x≤n≤x+x^δ)λ(n) = o(x^δ) for almost all x ∈ [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.https://authors.library.caltech.edu/records/4ynbt-66h97On the Typical Size and Cancelations Among the Coefficients of Some Modular Forms
https://resolver.caltech.edu/CaltechAUTHORS:20180614-155813169
Authors: {'items': [{'id': 'Luca-F', 'name': {'family': 'Luca', 'given': 'Florian'}}, {'id': 'Radziwiłł-M', 'name': {'family': 'Radziwill', 'given': 'Maksym'}}, {'id': 'Shparlinski-I-E', 'name': {'family': 'Shparlinski', 'given': 'Igor E.'}}]}
Year: 2018
DOI: 10.48550/arXiv.1308.6606
We obtain a nontrivial upper bound for almost all elements of the sequences of real numbers which are multiplicative and at the prime indices are distributed according to the Sato-Tate density. Examples of such sequences come from coefficients of several L-functions of elliptic curves and modular forms. In particular, we show that |τ(n)| ≤ n^(11/2) (log n)^((−1/2) + o(1)) for a set of n of asymptotic density 1, where τ(n) is the Ramanujan τ function while the standard argument yields log 2 instead of −1/2 in the power of the logarithm. Another consequence of our result is that in the number of representations of n by a binary quadratic form one has slightly more than square-root cancellations for almost all integers n.
In addition we obtain a central limit theorem for such sequences, assuming a weak hypothesis on the rate of convergence to the Sato--Tate law. For Fourier coefficients of primitive holomorphic cusp forms such a hypothesis is known conditionally assuming the automorphy of all symmetric powers of the form and seems to be within reach unconditionally using the currently established potential automorphy.https://authors.library.caltech.edu/records/75grr-81g33On large deviations of additive functions
https://resolver.caltech.edu/CaltechAUTHORS:20180614-141222705
Authors: {'items': [{'id': 'Radziwiłł-M', 'name': {'family': 'Radziwill', 'given': 'Maksym'}}]}
Year: 2018
DOI: 10.48550/arXiv.0909.5274
We prove that if two additive functions (from a certain class) take large values with roughly the same probability then they must be identical. The Kac-Kubilius model suggests that the distribution of values of a given additive function can be modeled by a sum of random variables. We show that the model is accurate (in a large deviation sense) when one is looking at values of the additive function around its mean, but fails, by a constant multiple, for large values of the additive function. We believe that this phenomenon arises, because the model breaks down for the values of the additive function on the "large" primes.
In the second part of the paper, we are motivated by a question of Elliott, to understand how much the distribution of values of the additive function on primes determines, and is determined by, the distribution of values of the additive function on all of the integers. For example, our main theorem, implies that a positive, strongly additive function is roughly Poisson distributed on the integers if and only if it is 1+o(1) or o(1) on almost all primes.https://authors.library.caltech.edu/records/57myw-kp031Limitations to mollifying ζ(s)
https://resolver.caltech.edu/CaltechAUTHORS:20180614-153722868
Authors: {'items': [{'id': 'Radziwiłł-M', 'name': {'family': 'Radziwiłł', 'given': 'Maksym'}}]}
Year: 2018
DOI: 10.48550/arXiv.1207.6583
We establish limitations to how well one can mollify the Riemann zeta-function on the critical line with mollifiers of arbitrary length. Our result gives a non-trivial lower bound for the contribution of the off-diagonal terms to mollified moments of ζ(s). On the Riemann Hypothesis, we establish a connection between the mollified moment and Montgomery's Pair Correlation Function.https://authors.library.caltech.edu/records/epn58-52t98Large deviations in Selberg's central limit theorem
https://resolver.caltech.edu/CaltechAUTHORS:20180614-141729646
Authors: {'items': [{'id': 'Radziwiłł-M', 'name': {'family': 'Radziwiłł', 'given': 'Maksym'}}]}
Year: 2018
DOI: 10.48550/arXiv.1108.5092
Following Selberg it is known that as T → ∞, [formula] uniformly for Δ ≤ (log log log T)^((1/2) - ε). We extend the range of Δ to Δ « (log log T)^((1/10) - ε). We also speculate on the size of the largest Δ for which the above normal approximation can hold and on the correct approximation beyond this point.https://authors.library.caltech.edu/records/v0d11-tj207A structure theorem in probabilistic number theory
https://resolver.caltech.edu/CaltechAUTHORS:20180614-142437720
Authors: {'items': [{'id': 'Radziwiłł-M', 'name': {'family': 'Radziwiłł', 'given': 'Maksym'}}]}
Year: 2018
DOI: 10.48550/arXiv.1109.0033
We prove that if two additive functions (from a certain class) take large values with roughly the same probability then they must be identical. This is a consequence of a structure theorem making clear the inter-relation between the distribution of an additive function on the integers, and its distribution on the primes.https://authors.library.caltech.edu/records/e1tfv-b5z32A converse to Halász's theorem
https://resolver.caltech.edu/CaltechAUTHORS:20180614-142647694
Authors: {'items': [{'id': 'Radziwiłł-M', 'name': {'family': 'Radziwiłł', 'given': 'Maksym'}}]}
Year: 2018
DOI: 10.48550/arXiv.1109.0037
We show that the distribution of large values of an additive function on the integers, and the distribution of values of the additive function on the primes are related to each other via a Levy Process. As a consequence we obtain a converse to an old theorem of Halasz. Halasz proved that if f is an strongly additive function with f (p) ∈ {0, 1}, then f is Poisson distributed on the integers. We prove, conversely, that if f is Poisson distributed on the integers then for most primes p, f(p) = o(1) or f(p) = 1 + o(1).https://authors.library.caltech.edu/records/5gnfr-3gg89Expansion, divisibility and parity
https://resolver.caltech.edu/CaltechAUTHORS:20210825-184543983
Authors: {'items': [{'id': 'Helfgott-Harald-Andrés', 'name': {'family': 'Helfgott', 'given': 'Harald Andrés'}}, {'id': 'Radziwiłł-M', 'name': {'family': 'Radziwiłł', 'given': 'Maksym'}}]}
Year: 2021
DOI: 10.48550/arXiv.2103.06853
Let P ⊂ [H₀,H] be a set of primes, where log H₀ ≥(log H)^(2/3+ϵ). Let L = Σ_(pϵP)1/p). Let N be such that log H ≤ (log N)^(1/2- ϵ). We show there exists a subset X⊂(N, 2N] of density close to 1 such that all the eigenvalues of the linear operator (A_|Xf)(n) = Σ/pϵP:p|n/n,n±pϵX f(n±p) – Σ/pϵP/n±pϵX f(n±p)/p are O(√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 Matomaki, Radziwill and Tao on the average of λ(n) in short intervals, we derive that 1/log x Σ/(n≤xλ(n)λ(n+1)/n = 0(1/√log log x), improving on a result of Tao's. We also prove that Σ_(Nhttps://authors.library.caltech.edu/records/nn2s3-8m749Triple correlation and long gaps in the spectrum of flat tori
https://resolver.caltech.edu/CaltechAUTHORS:20210825-184506223
Authors: {'items': [{'id': 'Aistleitner-Christoph', 'name': {'family': 'Aistleitner', 'given': 'Christoph'}}, {'id': 'Blomer-Valentin', 'name': {'family': 'Blomer', 'given': 'Valentin'}}, {'id': 'Radziwiłł-M', 'name': {'family': 'Radziwiłł', 'given': 'Maksym'}}]}
Year: 2021
DOI: 10.48550/arXiv.1809.07881v1
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.https://authors.library.caltech.edu/records/xh1a9-dhr45The Fyodorov-Hiary-Keating Conjecture. I.
https://resolver.caltech.edu/CaltechAUTHORS:20210825-184537116
Authors: {'items': [{'id': 'Arguin-Louis-Pierre', 'name': {'family': 'Arguin', 'given': 'Louis-Pierre'}}, {'id': 'Bourgade-Paul', 'name': {'family': 'Bourgade', 'given': 'Paul'}}, {'id': 'Radziwiłł-M', 'name': {'family': 'Radziwiłł', 'given': 'Maksym'}}]}
Year: 2021
DOI: 10.48550/arXiv.2007.00988
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 ≤ t ≤ 2T for which
max_(|h|≤1|) ζ(1/2 + it +ih)| > e^y log T/((log log T)^(3/4))
is bounded by Cye^(−2y) uniformly in y ≥ 1. This is expected to be optimal for y = O(√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.https://authors.library.caltech.edu/records/9wk6x-70098Prime number theorem for analytic skew products
https://resolver.caltech.edu/CaltechAUTHORS:20210825-184533695
Authors: {'items': [{'id': 'Kanigowski-Adam', 'name': {'family': 'Kanigowski', 'given': 'Adam'}}, {'id': 'Lemańczyk-Mariusz', 'name': {'family': 'Lemańczyk', 'given': 'Mariusz'}}, {'id': 'Radziwiłł-M', 'name': {'family': 'Radziwiłł', 'given': 'Maksym'}}]}
Year: 2021
DOI: 10.48550/arXiv.2004.01125
We establish a prime number theorem for all uniquely ergodic, analytic skew
products on the 2-torus T². More precisely, for every irrational
α and every 1-periodic real analytic g : R → R of
zero mean, let T_(α,g) : T² → T² be defined by (x,y) → x+α ,y+g(x)). We prove that if T_(α,g) is uniquely ergodic then, for every (x,y) ∈ T², the sequence {T^p_(α,g)(x,y)} is equidistributed on T² 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 T².https://authors.library.caltech.edu/records/8hknz-5mq06Optimal Small Scale Equidistribution of Lattice Points on the Sphere, Heegner Points, and Closed Geodesics
https://resolver.caltech.edu/CaltechAUTHORS:20210825-184526761
Authors: {'items': [{'id': 'Humphries-Peter', 'name': {'family': 'Humphries', 'given': 'Peter'}}, {'id': 'Radziwiłł-M', 'name': {'family': 'Radziwiłł', 'given': 'Maksym'}}]}
Year: 2021
DOI: 10.48550/arXiv.1910.01360
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 < 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.https://authors.library.caltech.edu/records/dd6b2-amf63One-level density estimates for Dirichlet L-functions with extended support
https://resolver.caltech.edu/CaltechAUTHORS:20210825-184530218
Authors: {'items': [{'id': 'Drappeau-Sary', 'name': {'family': 'Drappeau', 'given': 'Sary'}, 'orcid': '0000-0001-8544-8549'}, {'id': 'Pratt-Kyle', 'name': {'family': 'Pratt', 'given': 'Kyle'}, 'orcid': '0000-0001-6807-8021'}, {'id': 'Radziwiłł-M', 'name': {'family': 'Radziwiłł', 'given': 'Maksym'}}]}
Year: 2021
DOI: 10.48550/arXiv.2002.11968
We estimate the 1-level density of low-lying zeros of L(s,χ) with χ ranging over primitive Dirichlet characters of conductor ∈[Q/2,Q] and for test functions whose Fourier transform is supported in [−2−50/1093,2+50/1093]. Previously any extension of the support past the range [−2,2] was only known conditionally on deep conjectures about the distribution of primes in arithmetic progressions, beyond the reach of the Generalized Riemann Hypothesis (e.g Montgomery's conjecture). Our work provides the first example of a family of L-functions in which the support is unconditionally extended past the "trivial range" that follows from a simple application of the underlying trace formula (in this case orthogonality of characters). We also highlight consequences for non-vanishing of L(s,χ).https://authors.library.caltech.edu/records/pt5p6-t2109Level of distribution of unbalanced convolutions
https://resolver.caltech.edu/CaltechAUTHORS:20210825-184513041
Authors: {'items': [{'id': 'Fouvry Étienne-', 'name': {'family': 'Fouvry', 'given': 'Étienne'}}, {'id': 'Radziwiłł-M', 'name': {'family': 'Radziwiłł', 'given': 'Maksym'}}]}
Year: 2021
DOI: 10.48550/arXiv.1811.08672
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)^ε) integers then the resulting multiplicative convolution has (in a weak sense) level of distribution x^(1/2 + 1/66 − ε) as x goes to infinity. This dispersion estimate has a number of consequences for: the distribution of the kth divisor function to moduli x^(1/2 + 1/66 − ε) for any integer k ≥ 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 − ε) to moduli as large as x^(1−ε) and for the Brun-Titchmarsh theorem for almost all moduli q of size x^(1−ε), 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.https://authors.library.caltech.edu/records/eh492-jfg69Higher uniformity of bounded multiplicative functions in short intervals on average
https://resolver.caltech.edu/CaltechAUTHORS:20210825-184540549
Authors: {'items': [{'id': 'Matomäki-Kaisa', 'name': {'family': 'Matomäki', 'given': 'Kaisa'}}, {'id': 'Radziwiłł-M', 'name': {'family': 'Radziwiłł', 'given': 'Maksym'}}, {'id': 'Tao-Terence', 'name': {'family': 'Tao', 'given': 'Terence'}}, {'id': 'Teräväinen-Joni', 'name': {'family': 'Teräväinen', 'given': 'Joni'}, 'orcid': '0000-0001-6258-8004'}, {'id': 'Ziegler-Tamar', 'name': {'family': 'Ziegler', 'given': 'Tamar'}}]}
Year: 2021
DOI: 10.48550/arXiv.2007.15644
Let λ denote the Liouville function. We show that, as X → ∞,
∫^(2X)_X sup_(P(Y)∈ℝ[Y]deg(P)≤k) ∣ ∑_(x≤n≤x+H) λ(n)e(−P(n))∣ dx = o(XH)
for all fixed k and X^θ ≤ H ≤ X with 0 < θ < 1 fixed but arbitrarily small. Previously this was only established for k ≤ 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 F⎯(g(n)Γ). By the inverse theory for the Gowers norms this implies the higher order asymptotic uniformity result
∫^(2X)_X ‖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 + P₁(m),…,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 ≥ exp ((log X)^(5/8 + ε)), thus strengthening also previous work on the Fourier uniformity of the Liouville function.https://authors.library.caltech.edu/records/kc99k-vxt09Bias in cubic Gauss sums: Patterson's conjecture
https://resolver.caltech.edu/CaltechAUTHORS:20230227-232631259
Authors: {'items': [{'id': 'Dunn-Alexander-J', 'name': {'family': 'Dunn', 'given': 'Alexander'}, 'orcid': '0000-0003-1665-7114'}, {'id': 'Radziwiłł-M', 'name': {'family': 'Radziwiłł', 'given': 'Maksym'}}]}
Year: 2023
DOI: 10.48550/arXiv.2109.07463
Let W be a smooth test function with compact support in (0,∞). Conditional on the Generalized Riemann Hypothesis for Hecke L-functions over ℚ(ω), we prove that
_p≡1∑_((mod3)) 1/2‾√p⋅ₓ(∑_((modp)) e^((2πix³/p))W(p/X)∼(2π)^(2/3)/3Γ(2/3) ∫^∞_0 W(x)x^(−1/6) dx ⋅ X^(5/6)/logX,
as X→∞ and p runs over primes. This explains a well-known numerical bias in the distribution of cubic Gauss sums first observed by Kummer in 1846 and confirms (conditionally on the Generalized Riemann Hypothesis) a conjecture of Patterson from 1978.
There are two important byproducts of our proof. The first is an explicit level aspect Voronoi summation formula for cubic Gauss sums, extending computations of Patterson and Yoshimoto. Secondly, we show that Heath-Brown's cubic large sieve is sharp up to factors of X^(o(1)) under the Generalized Riemann Hypothesis. This disproves the popular belief that the cubic large sieve can be improved.
An important ingredient in our proof is a dispersion estimate for cubic Gauss sums. It can be interpreted as a cubic large sieve with correction by a non-trivial asymptotic main term. This estimate relies on the Generalized Riemann Hypothesis, and is one of the fundamental reasons why our result is conditional.https://authors.library.caltech.edu/records/gj1ev-0dk12