Disjointness of Möbius from asymptotically periodic functions

Wei, Fei

doi:10.4310/pamq.2022.v18.n3.a3

Cited by 2 publications

(2 citation statements)

References 52 publications

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“…for |𝑡| ⩽ 𝑥. Applying (46) or (47) with 𝜒 = 𝜒 1 𝜒 2 and 𝑡 = 𝑡 1 − 𝑡 2 for each (𝜒 1 , 𝑡 1 ), (𝜒 2 , 𝑡 2 ) ∈  𝑞, counted by 𝑆 2 , and handling the contributions from these as in (45), we find 𝑆 2 ≪ (𝜀 10 𝑁 2 𝑞, + 𝑁 𝑞, )𝑃.…”

Section: Key Propositionsmentioning

confidence: 95%

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Multiplicative functions in short arithmetic progressions

Klurman

Mangerel

Teräväinen

2023

Proceedings of London Math Soc

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We study for bounded multiplicative functions f$f$ sums of the form ∑n⩽xnbadbreak≡a0.33emfalse(mod0.33emqfalse)ffalse(nfalse),$$\begin{align*} \hspace*{7pc}\sum _{\substack{n\leqslant x\\ n\equiv a\ (\mathrm{mod}\ q)}}f(n), \end{align*}$$establishing that their variance over residue classes a0.33emfalse(mod0.33emqfalse)$a \ (\mathrm{mod}\ q)$ is small as soon as q=ofalse(xfalse)$q=o(x)$, for almost all moduli q$q$, with a nearly power‐saving exceptional set of q$q$. This improves and generalizes previous results of Hooley on Barban–Davenport–Halberstam type theorems for such f$f$, and moreover our exceptional set is essentially optimal unless one is able to make progress on certain well‐known conjectures. We are nevertheless able to prove stronger bounds for the number of the exceptional moduli q$q$ in the cases where q$q$ is restricted to be either smooth or prime, and conditionally on GRH we show that our variance estimate is valid for every q$q$. These results are special cases of a “hybrid result” that we establish that works for sums of f$f$ over almost all short intervals and arithmetic progressions simultaneously, thus generalizing the Matomäki–Radziwiłł theorem on multiplicative functions in short intervals. We also consider the maximal deviation of f$f$ over all residue classes a0.33emfalse(mod0.33emqfalse)$a\ (\mathrm{mod}\ q)$ in the square root range q⩽x1/2−ε$q\leqslant x^{1/2-\varepsilon }$, and show that it is small for “smooth‐supported” f$f$, again apart from a nearly power‐saving set of exceptional q$q$, thus providing a smaller exceptional set than what follows from Bombieri–Vinogradov type theorems. As an application of our methods, we consider Linnik‐type problems for products of exactly three primes, and in particular prove a ternary approximation to a conjecture of Erdős on representing every element of the multiplicative group double-struckZp×$\mathbb {Z}_p^{\times }$ as the product of two primes less than p$p$.

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Section: Key Propositionsmentioning

confidence: 95%

“…We can also specialize Corollary 1.6 to

f=\mu

and to the smaller range

q\leqslant x^{\varepsilon ^{200}}

to obtain a clean statement, which has recently been used in [47] to obtain applications to ergodic theory. Corollary Let

A\geqslant 1

be fixed.…”

Section: Main Theoremsmentioning

confidence: 99%

Multiplicative functions in short arithmetic progressions

Klurman

Mangerel

Teräväinen

2023

Proceedings of London Math Soc

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Sarnak's Möbius disjointness for dynamical systems with singular spectrum and dissection of Möbius flow

Abdalaoui

Nerurkar

2020

Preprint

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It is shown that Sarnak's Möbius orthogonality conjecture is fulfilled for the compact metric dynamical systems for which every invariant measure has singular spectra. This is accomplished by first establishing a special case of Chowla conjecture which gives a correlation between the Möbius function and its square. Then a computation of W. Veech, followed by an argument using the notion of 'affinity between measures', (or the so called 'Hellinger method'), completes the proof. We further present an unpublished theorem of Veech which is closely related to our main result. This theorem asserts, if for any probability measure in the closure of the Cesaro averages of the Dirac measure on the shift of the Möbius function, the first projection is in the orthocomplement of its Pinsker algebra then Sarnak Möbius disjointness conjecture holds. Among other consequences, we obtain a simple proof of Matomäki-Radziwi ll-Tao's theorem and Matomäki-Radziwi ll's theorem on the correlations of order two of the Liouville function.

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Disjointness of Möbius from asymptotically periodic functions

Cited by 2 publications

References 52 publications

Multiplicative functions in short arithmetic progressions

Multiplicative functions in short arithmetic progressions

Sarnak's Möbius disjointness for dynamical systems with singular spectrum and dissection of Möbius flow

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