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SUMMARY:Asher Roberts (St. Joseph's University New York)
DTSTART:20250521T180000Z
DTEND:20250521T182500Z
DTSTAMP:20260423T041655Z
UID:CANT2025/33
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/CANT2025/33/
 ">Large deviations of Selberg's central limit theorem on RH</a>\nby Asher 
 Roberts (St. Joseph's University New York) as part of Combinatorial and ad
 ditive number theory (CANT 2025)\n\nLecture held in CUNY Graduate Center -
  Science Center (4th floor).\n\nAbstract\nAssuming the Riemann hypothesis\
 , we show that for $k>0$ and $V\\sim k\\log\\log T$\,\n     \\[\n     \\fr
 ac{1}{T}\\operatorname{meas}\\bigg\\{t\\in[T\,2T]: \\log |\\zeta(1/2+{\\rm
  i} t)|>V\\bigg\\}\\leq C_k \\frac{e^{-V^2/\\log\\log T}}{\\sqrt{\\log\\lo
 g T}}.\n     \\]\n     This shows that Selberg's central limit theorem per
 sists in the large deviation regime. As a corollary\, we recover the resul
 t of Soundararajan and of Harper on the moments of $\\zeta$. This directly
  implies the sharp moment bounds of Soundararajan and Harper\, i.e.\,\n   
   \\[\n     \\frac{1}{T}\\int_T^{2T}|\\zeta(1/2+{\\rm i} t)|{\\rm d}t\\leq
  C_k (\\log T)^{k^2}.\n     \\]\n     This is joint work with Louis-Pierre
  Arguin (Oxford University) and Emma Bailey (University of Bristol).\n
LOCATION:https://researchseminars.org/talk/CANT2025/33/
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