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SUMMARY:Alex Walker
DTSTART:20220202T160000Z
DTEND:20220202T170000Z
DTSTAMP:20260418T070125Z
UID:LNTS/60
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/LNTS/60/">Th
 e Average Number of Divisors of the Output of a Quadratic Polynomial</a>\n
 by Alex Walker as part of London number theory seminar\n\n\nAbstract\nLet 
 $d(n)$ count the number of divisors of $n$. In 1963\, Hooley studied parti
 al sums $n < X$ of $d(n^2+h)$ and showed that the result was asymptotic to
  $c X \\log X + c' X + O(X^{8/9})$ as $X$ tends to infinity (assuming $h$ 
 not a negative square). In other words\, the irreducible polynomial $Q(x) 
 = x^2 + h$ has outputs with\, on average\, $\\sim \\log x$ many divisors. 
 Hooley's error bound was improved by Bykoskii in 1987 to $O(X^{2/3})$ usin
 g the spectral theory of automorphic forms. This talk describes a new proo
 f of Bykovskii's result in a new framework\, now using Dirichlet series an
 d automorphic forms of half-integral weight. This new framework has limita
 tions but is also quite flexible. To demonstrate this\, we develop in tand
 em counts for the average number of divisors of $Q(x\,y) = x^2+y^2+h$ for 
 $x^2+y^2+h < X$.\n
LOCATION:https://researchseminars.org/talk/LNTS/60/
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