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SUMMARY:Jeffrey C. Lagarias (University of Michigan)
DTSTART:20200601T203000Z
DTEND:20200601T205500Z
DTSTAMP:20260423T011223Z
UID:CANT2020/14
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/CANT2020/14/
 ">Partial factorizations of products of binomial coefficients</a>\nby Jeff
 rey C. Lagarias (University of Michigan) as part of Combinatorial and addi
 tive number theory (CANT 2021)\n\n\nAbstract\nLet $G_n$ denote the product
  of the binomial coefficients in the $n$-th row of \nPascal's triangle.  T
 hen $\\log G_n$ is asymptotic to $\\frac{1}{2}n^2$ as $n \\to \\infty$.\nL
 et $G(n\,x)$ denote the product of the maximal prime powers of all $p \\le
  x$ dividing $G_n$. \nWe determine asymptotics of $\\log G(n\, \\alpha n) 
 \\sim f(\\alpha)n^2$ as $n \\to \\infty$\,\nwith error term. Here \n\\[\nf
 (\\alpha) = \\frac{1}{2}   -\\alpha \\left\\lfloor \\frac{1}{\\alpha} \\ri
 ght\\rfloor\n+ \\frac{1}{2} \\alpha^2  \\left\\lfloor \\frac{1}{\\alpha}\\
 right\\rfloor^2 + \\frac{1}{2} \\alpha^2  \\left\\lfloor \\frac{1}{\\alpha
 }\n \\right\\rfloor \n\\]\nfor $0< \\alpha \\le 1$.\n The result is based 
 on  analysis of associated radix expansion statistics $A(n\,x)$ and $B(n\,
 x)$.\n The estimates relate to prime number theory\, and vice versa.\n\nJo
 int work with Lara Du.\n
LOCATION:https://researchseminars.org/talk/CANT2020/14/
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