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SUMMARY:Fourier uniformity of the Liouville function in shorter intervals
DTSTART:20260112T143000Z
DTEND:20260112T153000Z
DTSTAMP:20260306T212100Z
UID:indico-event-14774@indico.math.cnrs.fr
CONTACT:regis.de-la-breteche@imj-prg.fr
DESCRIPTION:Speakers: Cédric Pilatte (University of Oxford\, Royaume-Uni)
 \n\nIn this talk\, we consider the pseudo-random behaviour of the Liouvill
 e function in short intervals. Let $H(X)$ be an increasing function. We sa
 y that the Liouville function is <em>Fourier-uniform</em> in almost all in
 tervals of length $H = H(X)$ at scale $X$ if $$\\sum_{X < x < 2X} \\sup_{\
 \alpha \\in \\mathbb{R}}\\\, \\left\\lvert \\sum_{x < n < x+H} \\lambda(n)
  e(n\\alpha) \\right\\rvert = o(HX)$$ as $X \\to \\infty$.The Fourier unif
 ormity conjecture predicts that this estimate holds for any function $H(X)
 $ going to infinity with $X$ (arbitrarily slowly). This would generalise t
 he Matomäki-Radziwiłł theorem\, which establishes this when $\\alpha$ i
 s fixed and independent of $x$.\nIn 2023\, Walsh showed that $\\lambda$ is
  Fourier-uniform in almost all intervals of length $\\exp((\\log X)^{1/2+\
 \varepsilon})$. We improve this to $\\exp((\\log X)^{2/5+\\varepsilon})$\,
  a natural barrier for this problem.\n\nhttps://indico.math.cnrs.fr/event/
 14774/
LOCATION:Salle Yvette Cauchois (IHP - Bâtiment Perrin)
URL:https://indico.math.cnrs.fr/event/14774/
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