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SUMMARY:Austin Abraham Cramer (Penn State)
DTSTART:20260716T203000Z
DTEND:20260716T205500Z
DTSTAMP:20260805T030859Z
UID:CANT2026/93
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/CANT2026/93/
 ">Lattice points under polynomial curves</a>\nby Austin Abraham Cramer (Pe
 nn State) as part of Combinatorial and additive number theory seminar (CAN
 T 2026)\n\nLecture held in Science Center in the CUNY Graduate Center (4th
  floor).\n\nAbstract\nThis presentation is concerned with various analogs 
 of the Gauss circle and Dirichlet  divisor problems. For a given polynomia
 l $f(x_1\,\\ldots\,x_s)$ with integer coefficients\, let  $r_f(n) = \\#\\{
 (x_1\,\\ldots\,x_s) \\in \\mathbb{N}^s: f(x_1\,\\ldots\,x_s) = n\\}$. Usin
 g van der Corput's method to control error terms\, we obtain asymptotic fo
 rmulas for averages of this function of the form $\\sum_{n \\leq N} r_f(n)
 $ with various choices of $f$. When $f(x_1\,x_2)$ is a polynomial in two v
 ariables\, this counts the lattice points in the first quadrant bounded by
  the curve $f(x_1\,x_2) = N$. In this setting\, we demonstrate how to obta
 in asymptotics for various curves of low degree as well as how the method 
 can be extended to apply to some simple hypersurfaces bounding points in $
 \\mathbb{N}^s$.\n
LOCATION:https://researchseminars.org/talk/CANT2026/93/
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