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SUMMARY:Taylor Daniels (Purdue University)
DTSTART:20250523T150000Z
DTEND:20250523T152500Z
DTSTAMP:20260423T041811Z
UID:CANT2025/62
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/CANT2025/62/
 ">Vanishing Legendre-$17$-signed partition numbers</a>\nby Taylor Daniels 
 (Purdue University) as part of Combinatorial and additive number theory (C
 ANT 2025)\n\nLecture held in CUNY Graduate Center - Science Center (4th fl
 oor).\n\nAbstract\nFor odd primes $p$ let $\\chi_p(r) := (\\frac{r}{p})$ d
 enote the Legendre symbol. With this\, the Legendre-signed partition numbe
 rs\, denoted $\\mathfrak{p}(n\,\\chi_{p})$\, are then defined to be the co
 efficients appearing in the series expansion \n$$\\prod_{r=1}^{p-1}\\prod_
 {m=0}^{\\infty}\\frac{1}{1-\\chi_{p}(r)q^{mp+r}} = 1 + \\sum_{n=1}^\\infty
  \\mathfrak{p}(n\,\\chi_{p})q^n.$$ \nIt is known that: (1) one has $\\math
 frak{p}(n\,\\chi_{5}) = 0$ for all $n \\equiv 2 \\\,(\\mathrm{mod}\\\,10)$
 \; and (2) the sequences $(\\mathfrak{p}(n\,\\chi_{p}))_{n \\geq 1}$ do no
 t have such a periodic vanishing whenever $p \\not\\equiv 1 \\\,(\\mathrm{
 mod}\\\,8)$ and $p \\neq 5$. In this talk we discuss the recent result tha
 t $\\mathfrak{p}(n\,\\chi_{17})$ vanishes only when the input $n$ is odd a
 nd $1-24n$ is congruent to a quartic residue $(\\mathrm{mod}\\\,17)$\, as 
 well as a similar vanishing in the sequence $(\\mathfrak{p}(n\,-\\chi_{17}
 ))_{n\\geq 1}$.\n
LOCATION:https://researchseminars.org/talk/CANT2025/62/
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