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SUMMARY:Glenn T. Bruda (University of Florida)
DTSTART:20260718T200000Z
DTEND:20260718T202500Z
DTSTAMP:20260805T054658Z
UID:CANT2026/91
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/CANT2026/91/
 ">Generalized polygonal number representations</a>\nby Glenn T. Bruda (Uni
 versity of Florida) as part of Combinatorial and additive number theory se
 minar (CANT 2026)\n\nLecture held in Science Center in the CUNY Graduate C
 enter (4th floor).\n\nAbstract\nLet $r_n^{{k}}(N)$ be the number of repres
 entations of $N$ as the sum of $n$ generalized $k$-gonal numbers and $r_n^
 {\\square}(N)$ be the number of representations of $N$ as the sum of $n$ s
 quares. By modifying the Heath-Brown circle method\, we prove a closed-for
 m asymptotic relation between $r_n^{{k}}(N)$ and $r_n^{\\square}(8(k-2)N+n
 (k-4)^2)$ for any $k\\geq3$ and any $n\\geq4$. Consequently\, we estimate 
 $\\sum_{N\\leq x}r_4^{{k}}(N)^2$ and\, via a result of Bringmann\, Jang\, 
 Kane\, and Tse\, prove a similar closed-form asymptotic relation between t
 he number $r_{4\,+}^{{k}}(N)$ of representations of $N$ as the sum of four
  ordinary $k$-gonal numbers and $r_4^{\\square}(8(k-2)N+n(k-4)^2)$. We als
 o show that if $4\\mid k$\, any strictly increasing infinite subsequence o
 n which $r_{4\,+}^{{k}}$ is bounded converges $2$-adically to $(k-4)^2/(4-
 2k)\\in\\mathbb{Z}_2$\, supplementing a result of Meng and Sun\, and if $4
 \\nmid k$\, there is no strictly increasing infinite subsequence on which 
 $r_{4\,+}^{{k}}$ is bounded.\n
LOCATION:https://researchseminars.org/talk/CANT2026/91/
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