Madhuparna Das

University of Exeter


On a Central Limit Theorem for Partition Functions Involving Ramanujan Sums


Linfoot Number Theory Seminar


20th November 2024, 11:00 am – 12:00 pm
Fry Building, 2.04


In this talk, we establish a central limit theorem for the random variable $\varpi_{n}$, which represents the number of summands in a random weighted partition of $n$ associated with the Ramanujan sum $c_q(n)$. The generating series for such random weighted partition is given by $\prod_{n\in\mathbb{N}}(1+uz^n)^{(c_q(n+1)-c_q(n))}$. This concept was first introduced by Erd\H{o}s and Lehner, where they studied the distribution of ratio $p(n,r)/p(n)$, with $r=(2\pi^2/3)^{-\tfrac{1}{2}}\sqrt{n}\log n+n\sqrt{n}$ as a function of $n$ and $p(n)$ denoting number of integer partitions of $n$. Additionally, we note how the proposed method can be extended to a class of positive integer-valued multiplicative functions subject to certain restrictions.





Organisers: Holly Green, Besfort Shala

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