Some open questions in analysis for Dirichlet
                    series

Saksman, Eero; Seip, Kristian

doi:10.1090/conm/679/13675

Cited by 11 publications

(6 citation statements)

References 66 publications

(98 reference statements)

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“…In particular, we find that E| n≤x f (n)| ≍ √ x (log log x) 1/4 , which proves Helson's [15] somewhat surprising conjecture that the first moment should be o( √ x), and disproves the counter-conjecture of Harper, Nikeghbali and Radziwi l l [13] (see also Conjecture 1 of Heap and Lindqvist [14]). Theorem 1 also implies a negative answer to the so-called embedding problem for Dirichlet polynomials (see Question 2 of [24], or Problem 2.1 of [25]) for all exponents 0 < 2q < 2. For it is easy to check using Riemann-Stieltjes integration that…”

Section: Introductionmentioning

confidence: 99%

“…, and observed that if this is true then a certain generalisation of Nehari's theorem from harmonic analysis is false. See Saksman and Seip's open problems paper [25] for a functional analysis perspective on Helson's conjecture and related questions. On the number theoretic side, it is well known that the Riemann Hypothesis is true if and only if n≤x µ(n) = O ǫ (x 1/2+ǫ ) for all positive ǫ.…”

Section: Introductionmentioning

confidence: 99%

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Moments of Random Multiplicative Functions, I: Low Moments, Better Than Squareroot Cancellation, and Critical Multiplicative Chaos

Harper

2020

Forum of Mathematics, Pi

138

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We determine the order of magnitude of E| n≤x f (n)| 2q , where f (n) is a Steinhaus or Rademacher random multiplicative function, and 0 ≤ q ≤ 1. In the Steinhaus case, this is equivalent to determining the order of lim T →∞In particular, we find that E| n≤x f (n)| ≍ √ x/(log log x) 1/4 . This proves a conjecture of Helson that one should have better than squareroot cancellation in the first moment, and disproves counter-conjectures of various other authors. We deduce some consequences for the distribution and large deviations of n≤x f (n).The proofs develop a connection between E| n≤x f (n)| 2q and the q-th moment of a critical, approximately Gaussian, multiplicative chaos, and then establish the required estimates for that. We include some general introductory discussion about critical multiplicative chaos to help readers unfamiliar with that area.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Moments of Random Multiplicative Functions, I: Low Moments, Better Than Squareroot Cancellation, and Critical Multiplicative Chaos

Harper

2020

Forum of Mathematics, Pi

138

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“…Nous avons essentiellement restreint la présente étude au cas

α = \frac{1}{2}

, qui se révèle être l'un des plus intéressants — voir le survol . Nous posons en conséquence

S false(scriptM false) = S_{1 / 2} false(scriptM false)

Γ false(N false) = {normalΓ}_{1 / 2} false(N false)

{normalΓ}^{*} false(N false) = {normalΓ}_{1 / 2}^{*} false(N false)

.…”

Section: Introduction Et éNoncé Des Résultatsunclassified

Sommes de Gál et applications

Bretèche

Tenenbaum

2018

Proc. London Math. Soc.

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We evaluate the asymptotic size of various sums of Gál type, in particular 0trueS(M):=∑m,n∈scriptM(m,n)[m,n],where scriptM is a finite set of integers. Elaborating on methods recently developed by Bondarenko and Seip, we obtain an asymptotic formula for log supfalse|scriptMfalse|=NS(M)/Nand derive new lower bounds for localized extreme values of the Riemann zeta‐function, for extremal values of some Dirichlet L‐functions at s=12, and for large character sums.

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“…There are also probabilistic and analytic motivations for studying them, see Saksman and Seip's open problems paper [18], for example. The introduction to the previous paper [8] in this sequence contains a more extensive discussion of some of these connections.…”

Section: Introductionmentioning

confidence: 99%

Moments of random multiplicative functions, II: High moments

Harper

2019

Alg. Number Th.

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We determine the order of magnitude of E| n≤x f (n)| 2q up to factors of size e O(q 2 ) , where f (n) is a Steinhaus or Rademacher random multiplicative function, for all real 1 ≤ q ≤ c log x log log x . In the Steinhaus case, we show that E| n≤x f (n)| 2q = e O(q 2 ) x q ( log x q log(2q) ) (q−1) 2 on this whole range. In the Rademacher case, we find a transition in the behaviour of the moments when q ≈ (1 + √ 5)/2, where the size starts to be dominated by "orthogonal" rather than "unitary" behaviour. We also deduce some consequences for the large deviations of n≤x f (n).The proofs use various tools, including hypercontractive inequalities, to connect E| n≤x f (n)| 2q with the q-th moment of an Euler product integral. When q is large, it is then fairly easy to analyse this integral. When q is close to 1 the analysis seems to require subtler arguments, including Doob's L p maximal inequality for martingales.where the constant C St (q) satisfies C St (q) = e −q 2 log q−q 2 log log q+O(q 2 ) for large q. For Rademacher random multiplicative f (n), when q = 1 we have that E| n≤x f (n)| 2 = n≤x, n squarefree 1 ∼ (6/π 2 )x, and for fixed integer q ≥ 2 we havewhere the constant C Rad (q) satisfies C Rad (q) = e −2q 2 log q−2q 2 log log q+O(q 2 ) for large q. As described in [9, 10], we actually have much more precise information about the constants C St (q), C Rad (q) (for example they factor into explicit "arithmetic" and "geometric" parts), but this will not be important for our purposes here.We would like to have information about E| n≤x f (n)| 2q when q ≥ 1 is not necessarily integral, and that allows q to vary as a function of x rather than being fixed.Regarding uniformity in q, Theorem 4.1 of Granville and Soundararajan [5] implies that for Steinhaus random multiplicative f (n), and uniformly for all large x and integers

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Some open questions in analysis for Dirichlet series

Abstract: We present some open problems and describe briefly some possible research directions in the emerging theory of Hardy spaces of Dirichlet series and their intimate counterparts, Hardy spaces on the infinite-dimensional torus. Links to number theory are emphasized throughout the paper.

Cited by 11 publications

References 66 publications

Moments of Random Multiplicative Functions, I: Low Moments, Better Than Squareroot Cancellation, and Critical Multiplicative Chaos

Moments of Random Multiplicative Functions, I: Low Moments, Better Than Squareroot Cancellation, and Critical Multiplicative Chaos

Sommes de Gál et applications

Moments of random multiplicative functions, II: High moments

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