A Note on Helson’s Conjecture on Moments of Random Multiplicative Functions

Harper, Adam J.; Nikeghbali, Ashkan; Radziwiłł, Maksym

doi:10.1007/978-3-319-22240-0_11

Cited by 36 publications

(56 citation statements)

References 21 publications

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“…This can be related to recent results of [11] (see also [12]), where the authors obtain the asymptotic behaviour of a Steinhaus random multiplicative function (basically a multiplicative random variable whose values at prime integers are uniformly distributed on the unit circle). This can be viewed as a random model for

θ_{q} (x, χ)

.…”

Section: Introductionsupporting

confidence: 62%

Shifted Moments of ‐functions and Moments of Theta Functions

Munsch¹

2016

Mathematika

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θ_{q} (x, χ)

.…”

Section: Introductionsupporting

confidence: 62%

Shifted Moments of ‐functions and Moments of Theta Functions

Munsch¹

2016

Mathematika

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“…This seems surprising from a number theoretic perspective, since one rarely expects to achieve better than squareroot cancellation. Exploring the conjecture, Weber [29] established various results; Bondarenko and Seip [4] proved a lower bound E| n≤x f (n)| ≫ √ x/(log x) δ for a certain small explicit δ > 0; and Harper, Nikeghbali and Radziwi l l [13] proved a stronger lower bound E| n≤x f (n)| ≫ √ x/(log log x) 3+o (1) . They also conjectured, in opposition to Helson's conjecture, that E| n≤x f (n)| 2q ∼ C(q)x q as x → ∞, for each fixed 0 ≤ q ≤ 1.…”

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

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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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“…In the following proof, we have adapted the method of [, Section 2] which relies crucially on Harper's work . We refer to [, pp. 150–152] for an illuminating outline of the method.…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…We have

\begin{matrix} true \\ left |0true \sum_{n \in {scriptS}_{x}} d_{α} false(n false) α^{- Ω (n)} z false(n false) n^{- 1 / 2 - 1 / log x - i t}| = \prod_{p ⩽ x} |0true 1 + \sum_{j = 1}^{\infty} d_{α} false(p^{j} false) α^{- j} z {(p)}^{j} p^{- j (1 / 2 + 1 / log x + i t)}| \\ left ≍ prefixexp (() Re (0true \sum_{p ⩽ x} z false(p false) p^{- 1 / 2 - 1 / log x - i t}) + \frac{1}{2 α} Re (0true \sum_{p ⩽ x} z {(p)}^{2} p^{- 1 - 2 / log x - 2 i t})) \end{matrix}

for all points of the configuration space

{false(z (p) false)}_{p ⩽ x}

. As in [, Lem. 1], we can modify the proof of [, Corollary 2] to show that

\begin{matrix} true \\ left sup_{\begin{matrix} 1 ⩽ t ⩽ 2 {(log log x)}^{2} \\ false| 1 - 2^{- i t} false| ⩾ 1 / 4 \end{matrix}} ((Re (0true \sum_{<...>})) \end{matrix}

…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…Using this terminology, we may think of Z N as a sum of random multiplicative functions. A probabilistic approach to problems of computing integral moments, based on this viewpoint, can be found in recent work of Harper [11,13] and Harper, Nikeghbali and Radziwi l l [14]. In many situations, both approaches apply equally well, but sometimes one viewpoint is more illuminating and profitable than the other, and occasionally it is useful to combine them.…”

Section: Introductionmentioning

confidence: 99%

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Pseudomoments of the Riemann zeta function

Bondarenko

Brevig

Saksman

et al. 2018

Bull. London Math. Soc.

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The 2kth pseudomoments of the Riemann zeta function ζ(s) are, following Conrey and Gamburd, the 2kth integral moments of the partial sums of ζ(s) on the critical line. For fixed k>1/2, these moments are known to grow like false(prefixlogNfalse)k2, where N is the length of the partial sum, but the true order of magnitude remains unknown when k⩽1/2. We deduce new Hardy–Littlewood inequalities and apply one of them to improve on an earlier asymptotic estimate when k→∞. In the case k<1/2, we consider pseudomoments of ζαfalse(sfalse) for α>1 and the question of whether the lower bound false(prefixlogNfalse)k2α2 known from earlier work yields the true growth rate. Using ideas from recent work of Harper, Nikeghbali and Radziwiłł and some probabilistic estimates of Harper, we obtain the somewhat unexpected result that these pseudomements are bounded below by logN to a power larger than k2α2 when k<1/e and N is sufficiently large.

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A Note on Helson’s Conjecture on Moments of Random Multiplicative Functions

Cited by 36 publications

References 21 publications

Shifted Moments of ‐functions and Moments of Theta Functions

Shifted Moments of ‐functions and Moments of Theta Functions

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

Pseudomoments of the Riemann zeta function

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