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Higher order uniformity of the Möbius function
Virtualconference
Authors : Matomäki, Kaisa (Author of the conference)
CIRM (Publisher )
11N25 - Distribution of integers with specified multiplicative constraints
11N64 - Other results on the distribution of values or the characterization of arithmetic functions
11B30 - Arithmetic combinatorics; higher degree uniformity
CIRM (Publisher )
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Abstract :
The Liouville function $\lambda(n)$ takes the value +1 or -1 depending on whether $n$ has an even or an odd number of prime factors. The Liouville function is closely related to the characteristic function of the primes and is believed to behave more-or-less randomly.
I will discuss my very recent work with Radziwill, Tao, Teräväinen, and Ziegler, where we show that, in almost all intervals of length $X^{\varepsilon}$, the Liouville function does not correlate with polynomial phases or more generally with nilsequences.
I will also discuss applications to superpolynomial number of sign patterns for the Liouville sequence and to a new averaged version of Chowla's conjecture.
Keywords : Liouville function; higher order uniformity; Chowla's conjecture
MSC Codes : I will discuss my very recent work with Radziwill, Tao, Teräväinen, and Ziegler, where we show that, in almost all intervals of length $X^{\varepsilon}$, the Liouville function does not correlate with polynomial phases or more generally with nilsequences.
I will also discuss applications to superpolynomial number of sign patterns for the Liouville sequence and to a new averaged version of Chowla's conjecture.
Keywords : Liouville function; higher order uniformity; Chowla's conjecture
11N25 - Distribution of integers with specified multiplicative constraints
11N64 - Other results on the distribution of values or the characterization of arithmetic functions
11B30 - Arithmetic combinatorics; higher degree uniformity
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Information on the Video
Film maker : Hennenfent, GuillaumeLanguage : English Available date : 29/09/2020 Conference Date : 10/09/2020 Subseries : Research talks arXiv category : Number Theory Mathematical Area(s) : Number Theory Format : MP4 (.mp4) - HD Video Time : 00:34:37 Targeted Audience : Researchers Download : https://videos.cirm-math.fr/2020-09-10_Matomaki.mp4 
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Information on the Event
Event Title : Additive Combinatorics / Combinatoire additiveEvent Organizers : Balandraud, Eric ; Dousse, Jehanne ; Girard, Benjamin ; Schmid, Wolfgang ; Tringali, Salvatore Dates : 07/09/2020 - 11/09/2020 Event Year : 2020 Event URL : https://conferences.cirm-math.fr/2228.html
Citation Data
DOI : 10.24350/CIRM.V.19653803Cite this video as: Matomäki, Kaisa (2020). Higher order uniformity of the Möbius function. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19653803 URI : http://dx.doi.org/10.24350/CIRM.V.19653803 |
See Also
- [Multi angle] Spectra of ultra-discrete limits / Author of the conference Zuk, Andrzej.
- [Virtualconference] An update on the sum-product problem in $\mathbb{R}$ / Author of the conference Stevens, Sophie.
- [Virtualconference] Zero-sum squares in bounded discrepancy {-1,1}-matrices / Author of the conference Montejano, Amanda.
- [Virtualconference] On the local distribution of the product of two shifted primes and application / Author of the conference Deshouillers, Jean-Marc.
Bibliography
- MATOMÄKI, Kaisa, RADZIWIŁŁ, Maksym, TAO, Terence, et al. Higher uniformity of bounded multiplicative functions in short intervals on average. arXiv preprint arXiv:2007.15644, 2020. - https://arxiv.org/abs/2007.15644
