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A Caltech Library Repository Feedhttp://www.rssboard.org/rss-specificationpython-feedgenenTue, 16 Apr 2024 15:51:46 +0000The 4.36th Moment of the Riemann Zeta-Function
https://resolver.caltech.edu/CaltechAUTHORS:20180614-114019101
Authors: {'items': [{'id': 'Radziwiłł-M', 'name': {'family': 'Radziwiłł', 'given': 'Maksym'}}]}
Year: 2011
DOI: 10.1093/imrn/rnr183
Conditionally on the Riemann Hypothesis, we obtain bounds of the correct order of magnitude for the 2kth moment of the Riemann zeta-function for all positive real k<2.181. This provides for the first time an upper bound of the correct order of magnitude for some k>2; the case of k=2 corresponds to a classical result of Ingham [11]. We prove our result by establishing a connection between moments with k>2 and the so-called twisted fourth moment. This allows us to appeal to a recent result of Hughes and Young [10]. Furthermore we obtain a point-wise bound for |ζ(1/2 + it)|^(2r) (with 0https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/vc099-aw182Continuous lower bounds for moments of zeta and L-functions
https://resolver.caltech.edu/CaltechAUTHORS:20180618-152037832
Authors: {'items': [{'id': 'Radziwiłł-M', 'name': {'family': 'Radziwiłł', 'given': 'Maksym'}}, {'id': 'Soundararajan-K', 'name': {'family': 'Soundararajan', 'given': 'Kannan'}}]}
Year: 2013
DOI: 10.1112/S0025579312001088
We obtain lower bounds of the correct order of magnitude for the 2kth moment of the Riemann zeta function for all k≥1. Previously such lower bounds were known only for rational values of k, with the bounds depending on the height of the rational number k. Our new bounds are continuous in k, and thus extend also to the case when k is irrational. The method is a refinement of an approach of Rudnick and Soundararajan, and applies also to moments of L-functions in families.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/1htqv-w4r88The Riemann-zeta function on vertical arithmetic progressions
https://resolver.caltech.edu/CaltechAUTHORS:20180614-113331440
Authors: {'items': [{'id': 'Li-Xiannan', 'name': {'family': 'Li', 'given': 'Xiannan'}}, {'id': 'Radziwiłł-M', 'name': {'family': 'Radziwiłł', 'given': 'Maksym'}}]}
Year: 2013
DOI: 10.1093/imrn/rnt197
We show that the twisted second moments of the Riemann zeta function averaged over the arithmetic progression 1/2 + i(an + b) with a>0, b real, exhibits a remarkable correspondence with the analogous continuous average and derive several consequences. For example, motivated by the linear independence conjecture, we show at least one third of the elements in the arithmetic progression an+b are not the ordinates of some zero of ζ(s) lying on the critical line. This improves on an earlier work of Martin and Ng. We then complement this result by producing large values of ζ(s) on arithmetic progressions which are of the same quality as the best Ω results currently known for ζ(1/2 + it) with t real.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/4rs4t-r1f60Simple zeros of primitive Dirichlet L-functions and the asymptotic large sieve
https://resolver.caltech.edu/CaltechAUTHORS:20180619-154848646
Authors: {'items': [{'id': 'Chandee-V', 'name': {'family': 'Chandee', 'given': 'Vorrapan'}}, {'id': 'Lee-Yoonbok', 'name': {'family': 'Lee', 'given': 'Yoonbok'}}, {'id': 'Liu-Sheng-Chi', 'name': {'family': 'Liu', 'given': 'Sheng-Chi'}}, {'id': 'Radziwiłł-M', 'name': {'family': 'Radziwiłł', 'given': 'Maksym'}}]}
Year: 2014
DOI: 10.1093/qmath/hat008
Assuming the generalized Riemann hypothesis, 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 the 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. As a corollary of our results, we obtain an asymptotic formula for a sum over characters similar to the one encountered in the Barban–Davenport–Halberstam Theorem.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/3q1a3-5nt48Gaps between zeros of ζ(s) and the distribution of zeros of ζ′(s)
https://resolver.caltech.edu/CaltechAUTHORS:20180612-085638847
Authors: {'items': [{'id': 'Radziwiłł-M', 'name': {'family': 'Radziwiłł', 'given': 'Maksym'}}]}
Year: 2014
DOI: 10.1016/j.aim.2014.02.010
We assume the Riemann Hypothesis in this paper. We settle a conjecture of Farmer and Ki in a stronger form. Roughly speaking we show that there is a positive proportion of small gaps between consecutive zeros of the zeta-function ζ(s) if and only if there is a positive proportion of zeros of ζ′(s) lying very closely to the half-line. Our work has applications to the Siegel zero problem. We provide a criterion for the non-existence of the Siegel zero, solely in terms of the distribution of the zeros of ζ′(s). Finally on the Riemann Hypothesis and the Pair Correlation Conjecture we obtain near optimal bounds for the number of zeros of ζ′(s) lying very closely to the half-line. Such bounds are relevant to a deeper understanding of Levinson's method, allowing us to place one-third of the zeros of the Riemann zeta-function on the half-line.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/a5j5c-71630A Note on Helson's Conjecture on Moments of Random Multiplicative Functions
https://resolver.caltech.edu/CaltechAUTHORS:20180612-151835443
Authors: {'items': [{'id': 'Harper-A-J', 'name': {'family': 'Harper', 'given': 'Adam J.'}}, {'id': 'Nikeghbali-A', 'name': {'family': 'Nikeghbali', 'given': 'Ashkan'}}, {'id': 'Radziwiłł-M', 'name': {'family': 'Radziwiłł', 'given': 'Maksym'}}]}
Year: 2015
DOI: 10.1007/978-3-319-22240-0_11
In this note we are interested in cancellations in sums of multiplicative functions. It is well known that M(x) := Σ n≤x µ(n) = 0(x^(1/2+ε) is equivalent to the Riemann Hypothesis. On the other hand, it is also a classical result that M(x) > x^(1/2+ε) for a sequence of arbitrarily large x. It is in fact conjectured that lim x→ ∞ M(x)/ √x(log log log x)^(5/4) = ±B for some constant B > 0 (see [21]).https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/38p2e-9qd44Counting arithmetic formulas
https://resolver.caltech.edu/CaltechAUTHORS:20180612-132553805
Authors: {'items': [{'id': 'Gnang-E-K', 'name': {'family': 'Gnang', 'given': 'Edinah K.'}}, {'id': 'Radziwiłł-M', 'name': {'family': 'Radziwiłł', 'given': 'Maksym'}}, {'id': 'Sanna-C', 'name': {'family': 'Sanna', 'given': 'Carlo'}}]}
Year: 2015
DOI: 10.1016/j.ejc.2015.01.007
An arithmetic formula is an expression involving only the constant 1, and the binary operations of addition and multiplication, with multiplication by 1 not allowed. We obtain an asymptotic formula for the number of arithmetic formulas evaluating to as goes to infinity, solving a conjecture of E.K. Gnang and D. Zeilberger. We give also an asymptotic formula for the number of arithmetic formulas evaluating to and using exactly multiplications. Finally we analyze three specific encodings for producing arithmetic formulas. For almost all integers , we compare the lengths of the arithmetic formulas for that each encoding produces with the length of the shortest formula for (which we estimate from below). We briefly discuss the time-space tradeoff offered by each.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/8jt9c-y1571An averaged form of Chowla's conjecture
https://resolver.caltech.edu/CaltechAUTHORS:20180614-152233982
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: 2015
DOI: 10.2140/ant.2015.9.2167
Let λ denote the Liouville function. A well-known conjecture of Chowla asserts that, for any distinct natural numbers h_1,..., h_k, one has [formula] as X → ∞. This conjecture remains unproven for any h_1,..., h_k with k ≥ 2. Using the recent results of Matomäki and Radziwiłł on mean values of multiplicative functions in short intervals, combined with an argument of Kátai and Bourgain, Sarnak, and Ziegler, we establish an averaged version of this conjecture, namely [forumla] X → ∞, whenever H = H(X) ≤ X goes to infinity as X → ∞ and k is fixed. Related to this, we give the exponential sum estimate [forumla] as X → ∞ uniformly for all α ∈ ℝ, with H as before. Our arguments in fact give quantitative bounds on the decay rate (roughly on the order of 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.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/xq01x-73d29Sign changes of Hecke eigenvalues
https://resolver.caltech.edu/CaltechAUTHORS:20180612-093441157
Authors: {'items': [{'id': 'Matomäki-K', 'name': {'family': 'Matomäki', 'given': 'Kaisa'}}, {'id': 'Radziwiłł-M', 'name': {'family': 'Radziwiłł', 'given': 'Maksym'}}]}
Year: 2015
DOI: 10.1007/s00039-015-0350-7
Let f be a holomorphic or Maass Hecke cusp form for the full modular group and write λ_f(n)for the corresponding Hecke eigenvalues. We are interested in the signs of those eigenvalues. In the holomorphic case, we show that for some positive constant δ and every large enough x, the sequence (λ_f(n))_(n≤x) has at least δx sign changes. Furthermore we show that half of non-zero λ_f(n) are positive and half are negative. In the Maass case, it is not yet known that the coefficients are non-lacunary, but our method is robust enough to show that on the relative set of non-zero coefficients there is a positive proportion of sign changes. In both cases previous lower bounds for the number of sign changes were of the form x^δ for some δ < 1.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/cfdw1-gt921Moments and distribution of central L-values of quadratic twists of elliptic curves
https://resolver.caltech.edu/CaltechAUTHORS:20180612-153105797
Authors: {'items': [{'id': 'Radziwiłł-M', 'name': {'family': 'Radziwiłł', 'given': 'Maksym'}}, {'id': 'Soundararajan-K', 'name': {'family': 'Soundararajan', 'given': 'K.'}}]}
Year: 2015
DOI: 10.1007/s00222-015-0582-z
We show that if one can compute a little more than a particular moment for some family of L-functions, then one has upper bounds of the conjectured order of magnitude for all smaller (positive, real) moments and a one-sided central limit theorem holds. We illustrate our method for the family of quadratic twists of an elliptic curve, obtaining sharp upper bounds for all moments below the first. We also establish a one sided central limit theorem supporting a conjecture of Keating and Snaith. Our work leads to a conjecture on the distribution of the order of the Tate-Shafarevich group for rank zero quadratic twists of an elliptic curve, and establishes the upper bound part of this conjecture (assuming the Birch-Swinnerton-Dyer conjecture).https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/jtg6r-et652Multiplicative functions in short intervals
https://resolver.caltech.edu/CaltechAUTHORS:20180612-125453636
Authors: {'items': [{'id': 'Matomäki-K', 'name': {'family': 'Matomäki', 'given': 'Kaisa'}}, {'id': 'Radziwiłł-M', 'name': {'family': 'Radziwiłł', 'given': 'Maksym'}}]}
Year: 2016
DOI: 10.4007/annals.2016.183.3.6
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)→∞ 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 xε-smooth numbers in intervals of the form [x,x+c(ε)√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 nontrivially 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 nonzero 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.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/1ga51-zar83Sign patterns of the Liouville and Möbius functions
https://resolver.caltech.edu/CaltechAUTHORS:20180614-151134685
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: 2016
DOI: 10.1017/fms.2016.6
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.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/64hyx-e1119Small gaps in the spectrum of the rectangular billiard
https://resolver.caltech.edu/CaltechAUTHORS:20180619-155911662
Authors: {'items': [{'id': 'Blomer-V', 'name': {'family': 'Blomer', 'given': 'Valentin'}}, {'id': 'Bourgain-J', 'name': {'family': 'Bourgain', 'given': 'Jean'}}, {'id': 'Radziwiłł-M', 'name': {'family': 'Radziwiłł', 'given': 'Maksym'}}, {'id': 'Rudnick-Z', 'name': {'family': 'Rudnick', 'given': 'Zeév'}}]}
Year: 2017
DOI: 10.48550/arXiv.1604.02413
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.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/ktf30-rhr64Refinements of Gál's theorem and applications
https://resolver.caltech.edu/CaltechAUTHORS:20180612-084828993
Authors: {'items': [{'id': 'Lewko-M', 'name': {'family': 'Lewko', 'given': 'Mark'}}, {'id': 'Radziwiłł-M', 'name': {'family': 'Radziwiłł', 'given': 'Maksym'}}]}
Year: 2017
DOI: 10.1016/j.aim.2016.09.006
We give a simple proof of a well-known theorem of Gál and of the recent related results of Aistleitner, Berkes and Seip [1] regarding the size of GCD sums. In fact, our method obtains the asymptotically sharp constant in Gál's theorem, which is new. Our approach also gives a transparent explanation of the relationship between the maximal size of the Riemann zeta function on vertical lines and bounds on GCD sums; a point which was previously unclear. Furthermore we obtain sharp bounds on the spectral norm of GCD matrices which settles a question raised in [2]. We use bounds for the spectral norm to show that series formed out of dilates of periodic functions of bounded variation converge almost everywhere if the coefficients of the series are in L^2(log log 1/L)^γ, with γ>2. This was previously known with γ>4, and is known to fail for γ<2. We also develop a sharp Carleson–Hunt-type theorem for functions of bounded variations which settles another question raised in [1]. Finally we obtain almost sure bounds for partial sums of dilates of periodic functions of bounded variations improving [1].https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/xfqf0-ca123The mean square of the product of the Riemann zeta-function with Dirichlet polynomials
https://resolver.caltech.edu/CaltechAUTHORS:20180614-154126641
Authors: {'items': [{'id': 'Bettin-S', 'name': {'family': 'Bettin', 'given': 'Sandro'}}, {'id': 'Chandee-V', 'name': {'family': 'Chandee', 'given': 'Vorrapan'}}, {'id': 'Radziwiłł-M', 'name': {'family': 'Radziwiłł', 'given': 'Maksym'}}]}
Year: 2017
DOI: 10.1515/crelle-2014-0133
Improving earlier work of Balasubramanian, Conrey and Heath-Brown [1], we obtain an asymptotic formula for the mean-square of the Riemann zeta-function times an arbitrary Dirichlet polynomial of length T^((1/2) + δ), with δ = 0.01515…. As an application we obtain an upper bound of the correct order of magnitude for the third moment of the Riemann zeta-function. We also refine previous work of Deshouillers and Iwaniec [8], obtaining asymptotic estimates in place of bounds. Using the work of Watt [19], we compute the mean-square of the Riemann zeta-function times a Dirichlet polynomial of length going up to T^(3/4) provided that the Dirichlet polynomial assumes a special shape. Finally, we exhibit a conjectural estimate for trilinear sums of Kloosterman fractions which implies the Lindelöf Hypothesis.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/vgem6-1mj33Another application of Linnik's dispersion method
https://resolver.caltech.edu/CaltechAUTHORS:20210825-184516487
Authors: {'items': [{'id': 'Fouvry-Étienne', 'name': {'family': 'Fouvry', 'given': 'Étienne'}}, {'id': 'Radziwiłł-M', 'name': {'family': 'Radziwiłł', 'given': 'Maksym'}}]}
Year: 2018
DOI: 10.22405/2226-8383-2018-19-3-148-163
Let be α_m and β_n---two sequences of real numbers supported on
segments [M,2M] and [N,2N], where M = X^(1/2 − δ) and N = X^(1/2 + δ)... We prove the
existence of such a constant δ₀ that the multiplicative convolution
α_m and β_n has a distribution level 1/2 + δ - ε (in a weak sense),
if only 0 ⩽ δ < δ₀, subsequence β_n is
a Siegel-Walvis sequence, and both sequences α_m and β_n
are bounded from above by the divisor function.
Our result, therefore, is the overall variance estimate
for "short", type II sums. The proof makes essential use of Linnik's dispersion method
and recent estimates for trilinear sums with Kloosterman fractions due to Bettin and Chandy.
We will also focus on the application of this result to the Titchmarsh divisor problem.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/t0v91-a7b43Selberg's central limit theorem for log |ζ(1/2+it)|
https://resolver.caltech.edu/CaltechAUTHORS:20180613-111022134
Authors: {'items': [{'id': 'Radziwiłł-M', 'name': {'family': 'Radziwiłł', 'given': 'Maksym'}}, {'id': 'Soundararajan-K', 'name': {'family': 'Soundararajan', 'given': 'Kannan'}}]}
Year: 2018
DOI: 10.4171/LEM/63-1/2-1
We present a new and simple proof of Selberg's central limit theorem, according to which the logarithm of the Riemann zeta-function at height t is approximately normally distributed with mean 0 and variance 1/2 log log t.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/edb2c-a9g38Maximum 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.eduhttps://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.eduhttps://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.eduhttps://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.eduhttps://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.eduhttps://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.eduhttps://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.eduhttps://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.eduhttps://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.eduhttps://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.eduhttps://authors.library.caltech.edu/records/5gnfr-3gg89Small scale distribution of zeros and mass of modular forms
https://resolver.caltech.edu/CaltechAUTHORS:20180614-154847306
Authors: {'items': [{'id': 'Lester-S', 'name': {'family': 'Lester', 'given': 'Stephen'}, 'orcid': '0000-0003-1977-9205'}, {'id': 'Matomäki-K', 'name': {'family': 'Matomäki', 'given': 'Kaisa'}}, {'id': 'Radziwiłł-M', 'name': {'family': 'Radziwiłł', 'given': 'Maksym'}}]}
Year: 2018
DOI: 10.4171/JEMS/794
We study the behavior of zeros and mass of holomorphic Hecke cusp forms on SL_2(ℤ)∖ℍ at small scales. In particular, we examine the distribution of the zeros within hyperbolic balls whose radii shrink sufficiently slowly as k→∞. We show that the zeros equidistribute within such balls as k→∞ 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(ℤ)∖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.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/9zjg1-1p358Correlations of the von Mangoldt and higher divisor functions II. Divisor correlations in short ranges
https://resolver.caltech.edu/CaltechAUTHORS:20180612-090354076
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: 2019
DOI: 10.1007/s00208-018-01801-4
We study the problem of obtaining asymptotic formulas for the sums ∑_(Xhttps://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/5cj74-bnb98Discrepancy bounds for the distribution of the Riemann zeta-function and applications
https://resolver.caltech.edu/CaltechAUTHORS:20180614-135438432
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: 2019
DOI: 10.1007/s11854-019-0063-1
We investigate the distribution of the Riemann zeta-function on the line Re(s) = σ. For ½ < σ ≤ 1 we obtain an upper bound on the discrepancy between the distribution of ζ (s) and that of its random model, improving results of Harman and Matsumoto. Additionally, we examine the distribution of the extreme values of ζ (s) inside of the critical strip, strengthening a previous result of the first author.
As an application of these results we obtain the first effective error term for the number of solutions to ζ (s) = a in a strip ½ < σ₁ < σ₂ < 1. Previously in the strip ½ < σ< 1 only an asymptotic estimate was available due to a result of Borchsenius and Jessen from 1948 and effective estimates were known only slightly to the left of the half-line, under the Riemann hypothesis (due to Selberg). In general our results are an improvement of the classical Bohr–Jessen framework and are also applicable to counting the zeros of the Epstein zeta-function.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/a7jdp-0rb12Sharp Upper Bounds for Fractional Moments of the Riemann Zeta Function
https://resolver.caltech.edu/CaltechAUTHORS:20200213-072834344
Authors: {'items': [{'id': 'Heap-W', 'name': {'family': 'Heap', 'given': 'Winston'}}, {'id': 'Radziwiłł-M', 'name': {'family': 'Radziwiłł', 'given': 'Maksym'}}, {'id': 'Soundararajan-K', 'name': {'family': 'Soundararajan', 'given': 'K.'}}]}
Year: 2019
DOI: 10.1093/qmathj/haz027
We establish sharp upper bounds for the 2kth moment of the Riemann zeta function on the critical line, for all real 0 ⩽ k ⩽ 2. This improves on earlier work of Ramachandra, Heath-Brown and Bettin–Chandee–Radziwiłł.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/j4hg2-9ng91Quantum Unique Ergodicity for half-integral weight automorphic forms
https://resolver.caltech.edu/CaltechAUTHORS:20180613-084101704
Authors: {'items': [{'id': 'Lester-S', 'name': {'family': 'Lester', 'given': 'Stephen'}, 'orcid': '0000-0003-1977-9205'}, {'id': 'Radziwiłł-M', 'name': {'family': 'Radziwiłł', 'given': 'Maksym'}}]}
Year: 2020
DOI: 10.1215/00127094-2019-0040
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 Γ₀(4) lying in Kohnen's plus subspace and for half-integral weight Hecke Maaβ cusp forms for Γ₀(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 Γ₀(4)∖H as the weight tends to infinity.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/gh1v7-mja51Fourier uniformity of bounded multiplicative functions in short intervals on average
https://resolver.caltech.edu/CaltechAUTHORS:20190927-105826819
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: 2020
DOI: 10.1007/s00222-019-00926-w
Let λ denote the Liouville function. We show that as X→∞,
∫^(2X)X supα∣∑x< n ≤ x+H λ(n)e(−αn)∣dx = o(XH) for all H ≥ 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.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/jnmcy-4tc61Limiting distribution of eigenvalues in the large sieve matrix
https://resolver.caltech.edu/CaltechAUTHORS:20180612-154327663
Authors: {'items': [{'id': 'Boca-F-P', 'name': {'family': 'Boca', 'given': 'Florin P.'}}, {'id': 'Radziwiłł-M', 'name': {'family': 'Radziwiłł', 'given': 'Maksym'}}]}
Year: 2020
DOI: 10.4171/JEMS/965
The large sieve inequality is equivalent to the bound λ₁ ≤ N+Q²−1 for the largest eigenvalue λ₁ of the N by N matrix A⋆A, naturally associated to the positive definite quadratic form arising in the inequality. For arithmetic applications the most interesting range is N≍Q². Based on his numerical data Ramaré conjectured that when N ∼ αQ² as Q → ∞ for some finite positive constant α, the limiting distribution of the eigenvalues of A⋆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⋆A as Q → ∞. 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.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/8msv2-qre36A quadratic divisor problem and moments of the Riemann zeta-function
https://resolver.caltech.edu/CaltechAUTHORS:20180613-075002928
Authors: {'items': [{'id': 'Bettin-Sandro', 'name': {'family': 'Bettin', 'given': 'Sandro'}}, {'id': 'Bui-Hung-M', 'name': {'family': 'Bui', 'given': 'Hung M.'}}, {'id': 'Li-Xiannan', 'name': {'family': 'Li', 'given': 'Xiannan'}}, {'id': 'Radziwiłł-M', 'name': {'family': 'Radziwiłł', 'given': 'Maksym'}}]}
Year: 2020
DOI: 10.4171/JEMS/999
We estimate asymptotically the fourth moment of the Riemann zeta-function twisted by a Dirichlet polynomial of length T 1/4^(−ε). 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 1/11^(−ε) 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 am₁m₂−bn₁n₂=h is summed with smooth averaging on the variables m₁,m₂,n₁,n₂,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 m₁,m₂,n₁,n₂,h right away in the quadratic divisor problem considerably simplifies the combinatorics of the main terms in the twisted fourth moment.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/12qxg-bfv20On the variance of squarefree integers in short intervals and arithmetic progressions
https://resolver.caltech.edu/CaltechAUTHORS:20210405-105931980
Authors: {'items': [{'id': 'Gorodetsky-Ofir', 'name': {'family': 'Gorodetsky', 'given': 'Ofir'}}, {'id': 'Matomäki-Kaisa', 'name': {'family': 'Matomäki', 'given': 'Kaisa'}}, {'id': 'Radziwiłł-M', 'name': {'family': 'Radziwiłł', 'given': 'Maksym'}}, {'id': 'Rodgers-Brad', 'name': {'family': 'Rodgers', 'given': 'Brad'}}]}
Year: 2021
DOI: 10.1007/s00039-021-00557-5
We evaluate asymptotically the variance of the number of squarefree integers up to x in short intervals of length H < x^(6/11−ε) and the variance of the number of squarefree integers up to x in arithmetic progressions modulo q with q > x^(5/11+ε). 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−ε) and q > x^(1/3+ε). Furthermore we show that obtaining a bound sharp up to factors of H^ε in the full range H < x^(1−ε) is equivalent to the Riemann Hypothesis. These results improve on a result of Hall (Mathematika 29(1):7–17, 1982) for short intervals, and earlier results of Warlimont, Vaughan, Blomer, Nunes and Le Boudec in the case of arithmetic progressions.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/q07cr-67626Signs of Fourier coefficients of half-integral weight modular forms
https://resolver.caltech.edu/CaltechAUTHORS:20210122-132850144
Authors: {'items': [{'id': 'Lester-Stephen', 'name': {'family': 'Lester', 'given': 'Stephen'}, 'orcid': '0000-0003-1977-9205'}, {'id': 'Radziwiłł-M', 'name': {'family': 'Radziwiłł', 'given': 'Maksym'}}]}
Year: 2021
DOI: 10.1007/s00208-020-02123-0
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 nth Fourier coefficient of g, normalized so that c(n) is real for all n≥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 4N with N odd, square-free.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/qa5h2-zts60A Note on the Dimension of the Largest Simple Hecke Submodule
https://resolver.caltech.edu/CaltechAUTHORS:20210602-105818575
Authors: {'items': [{'id': 'Bettin-Sandro', 'name': {'family': 'Bettin', 'given': 'Sandro'}}, {'id': 'Perret-Gentil-Corentin', 'name': {'family': 'Perret-Gentil', 'given': 'Corentin'}}, {'id': 'Radziwiłł-M', 'name': {'family': 'Radziwiłł', 'given': 'Maksym'}}]}
Year: 2021
DOI: 10.1093/imrn/rny287
For k ≥ 2 even, let d_(k,N) denote the dimension of the largest simple Hecke submodule of S_k(Γ₀ (N);Q)^(new). We show, using a simple analytic method, that d_(k,N) ≫ 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(Γ₀ (N),χ), with k ≥ 2 an integer and χ an arbitrary nebentypus.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/6epff-cqn23Rigidity in dynamics and Möbius disjointness
https://resolver.caltech.edu/CaltechAUTHORS:20210826-205458728
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: 10.4064/fm931-11-2020
Let (X,T) be a topological dynamical system. We show that if each invariant measure of (X,T) gives rise to a measure-theoretic dynamical system that is either: a. rigid along a sequence of "bounded prime volume" or b. admits a polynomial rate of rigidity on a linearly dense subset in C(X), then (X,T) satisfies Sarnak's conjecture on Möbius disjointness. We show that the same conclusion also holds if there are countably many invariant ergodic measures, and each of them satisfies a. or b. This recovers some earlier results and implies Sarnak's conjecture in the following new cases: for almost every interval exchange map of d intervals with d ≥ 2, for C^(2+ϵ)-smooth skew products over rotations and C^(2+ϵ)-smooth flows (without fixed points) on the torus. In particular, these are improvements of earlier results of respectively Chaika-Eskin, Wang and Huang-Wang-Ye. We also discuss some purely arithmetic consequences for the Liouville function.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/n9dyy-w3211Expansion, 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.eduhttps://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.eduhttps://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.eduhttps://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.eduhttps://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.eduhttps://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.eduhttps://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.eduhttps://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.eduhttps://authors.library.caltech.edu/records/kc99k-vxt09Moments of the Riemann zeta function on short intervals of the critical line
https://resolver.caltech.edu/CaltechAUTHORS:20210825-184519912
Authors: {'items': [{'id': 'Arguin-Louis-Pierre', 'name': {'family': 'Arguin', 'given': 'Louis-Pierre'}}, {'id': 'Ouimet-Frédéric', 'name': {'family': 'Ouimet', 'given': 'Frédéric'}, 'orcid': '0000-0001-7933-5265'}, {'id': 'Radziwiłł-M', 'name': {'family': 'Radziwiłł', 'given': 'Maksym'}}]}
Year: 2021
DOI: 10.1214/21-AOP1524
We show that as T→∞, for all t∈[T,2T] outside of a set of measure o(T),
∫^((log T)^θ⁰)_(−(log T)^θ) |ζ(1/2 + it + ih)|^β dh =(log T)^(f_θ(β) + 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 ∈ [T,2T] outside a set of measure o(T),
max_(|h| ≤ (log T)θ) |ζ(1/2 + it + ih)| = (log T)^(m(θ) + o(1)),
for some explicit m(θ). This generalizes earlier results of Najnudel (Probab. Theory Related Fields 172 (2018) 387–452) and Arguin et al. (Comm. Pure Appl. Math. 72 (2019) 500–535) for θ = 0. The proofs are unconditional, except for the upper bounds when θ > 3, where the Riemann hypothesis is assumed.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/b8ghv-6p743Level of distribution of unbalanced convolutions
https://resolver.caltech.edu/CaltechAUTHORS:20230301-701033500.5
Authors: {'items': [{'id': 'Fouvry-Étienne', 'name': {'family': 'Fouvry', 'given': 'Étienne'}}, {'id': 'Radziwiłł-M', 'name': {'family': 'Radziwill', 'given': 'Maksym'}}]}
Year: 2022
DOI: 10.24033/asens.2501
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.eduhttps://authors.library.caltech.edu/records/dfes3-qak12Singmaster's Conjecture In The Interior Of Pascal's Triangle
https://resolver.caltech.edu/CaltechAUTHORS:20210825-184547403
Authors: {'items': [{'id': 'Matomäki-Kaisa', 'name': {'family': 'Matomäki', 'given': 'Kaisa'}}, {'id': 'Radziwiłł-M', 'name': {'family': 'Radziwiłł', 'given': 'Maksym'}}, {'id': 'Shao-Xuancheng', 'name': {'family': 'Shao', 'given': 'Xuancheng'}}, {'id': 'Tao-Terence', 'name': {'family': 'Tao', 'given': 'Terence'}, 'orcid': '0000-0002-0140-7641'}, {'id': 'Teräväinen-Joni', 'name': {'family': 'Teräväinen', 'given': 'Joni'}, 'orcid': '0000-0001-6258-8004'}]}
Year: 2022
DOI: 10.1093/qmath/haac006
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 ≥ 2, the number of solutions to the equation (n m) = t for natural numbers 1 ≤ m < n is bounded. In this paper we establish this result in the interior region exp(log^(2/3+ε)n) ≤ m ≤ n − exp(log^(2/3+ε)n) for any fixed ɛ > 0. Indeed, when t is sufficiently large depending on ɛ, 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)…(n−m+1) denotes the falling factorial.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/56eba-q8x73Optimal Small Scale Equidistribution of Lattice Points on the Sphere, Heegner Points, and Closed Geodesics
https://resolver.caltech.edu/CaltechAUTHORS:20220920-390552800
Authors: {'items': [{'id': 'Humphries-Peter', 'name': {'family': 'Humphries', 'given': 'Peter'}}, {'id': 'Radziwiłł-M', 'name': {'family': 'Radziwiłł', 'given': 'Maksym'}}]}
Year: 2022
DOI: 10.1002/cpa.22076
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 weighting functions for annuli and for balls.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/71g7x-v3w27Bias 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.eduhttps://authors.library.caltech.edu/records/gj1ev-0dk12Higher uniformity of bounded multiplicative functions in short intervals on average
https://resolver.caltech.edu/CaltechAUTHORS:20230315-645491100.6
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'}, 'orcid': '0000-0002-0140-7641'}, {'id': 'Teräväinen-Joni', 'name': {'family': 'Teräväinen', 'given': 'Joni'}, 'orcid': '0000-0001-6258-8004'}, {'id': 'Ziegler-Tamar', 'name': {'family': 'Ziegler', 'given': 'Tamar'}}]}
Year: 2023
DOI: 10.4007/annals.2023.197.2.3
Let λ denote the Liouville function. We show that, as X → ∞, ∫^(2X)_X sup_(P(Y)∈R[Y]degP≤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 F̅(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+Pₖ(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.eduhttps://authors.library.caltech.edu/records/er3vb-rzz83One-level density estimates for Dirichlet L-functions with extended support
https://resolver.caltech.edu/CaltechAUTHORS:20230612-735582000.46
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: 2023
DOI: 10.2140/ant.2023.17.805
We estimate the 1-level density of low-lying zeros of L(s,χ) with χ ranging over primitive Dirichlet characters of conductor in [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.eduhttps://authors.library.caltech.edu/records/7qsc4-8xt58