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A central limit theorem for partitions involving generalised divisor function

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If you have a question about this talk, please contact Challenger Mishra .

We define an $f$-restricted partition $p_f(n,k)$ of fixed length $k$ 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 partitions when $f(n)=\sigma_r(n)$ denotes the generalised divisor function, defined as $\sigma_r(n)=\sum_{d|n}dr$ 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 $f(n)=\lfloor{n}{\alpha}\rfloor$ with $0<\alpha

\sum_{n=1}\frac{\sigma_r(n+1)}{ns},

for {Re}(s)>r+1. We study this problem employing the identity involving the Ramanujan sum.

Furthermore, we analyse the Euler product arising from the above Dirichlet series by adopting the argument of Alkan, Ledoan and Zaharescu.

Zoom: https://cl-cam-ac-uk.zoom.us/j/6590822098?pwd=VTBuUXRXN29qMDF4TGpaaEhFaytQQT09

This talk is part of the Mathematics and Computation series.

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