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SUMMARY:A central limit theorem for partitions involving generalised divis
 or function - Madhuparna Das\, Dept of Mathematics\, University of Exeter
DTSTART:20251015T140000Z
DTEND:20251015T150000Z
UID:TALK239185@talks.cam.ac.uk
CONTACT:Challenger Mishra
DESCRIPTION:We define an $f$-restricted partition $p_f(n\,k)$ of fixed len
 gth $k$ given by a bivariate generating series.  In this talk\, we outline
  the proof of a central limit theorem for the number of summands in such p
 artitions when $f(n)=\\sigma_r(n)$ denotes the generalised divisor functio
 n\, defined as $\\sigma_r(n)=\\sum_{d|n}d^r$ for integer $r\\geq 2$. This 
 can be considered as a generalisation of the work of Lipnik\, Madritsch\, 
 and Tichy\, who previously studied this problem for $f(n)=\\lfloor{n}^{\\a
 lpha}\\rfloor$ with $0<\\alpha<1$. A key element of our proof relies on th
 e analytic behaviour of the Dirichlet series\n\n\n\\sum_{n=1}^{\\infty}\\f
 rac{\\sigma_r(n+1)}{n^s}\,    \n\nfor {Re}(s)>r+1. We study this problem e
 mploying the identity involving the Ramanujan sum. \n\nFurthermore\, we an
 alyse the Euler product arising from the above Dirichlet series by adoptin
 g the argument of Alkan\, Ledoan and Zaharescu.\n\nZoom: https://cl-cam-ac
 -uk.zoom.us/j/6590822098?pwd=VTBuUXRXN29qMDF4TGpaaEhFaytQQT09
LOCATION:Computer Lab\, FW09 and Online (link in abstract)
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