Answer to Question 1 in Zeilbergers’ ‘Fractional Counting of Partitions’ 1

This is a short solution of Question 1 posed in the article Fractional Counting of Partitions

by Doron and Noam Zeilberger

   See http://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/fcp.html and

https://arxiv.org/abs/1810.12701 . The first solution was given by Will Sawin, and

is included as an appendix in the current version of their paper.

If C := lim \frac{b(n)}{n} exists then C = e^{−γ}.

Given ϵ > 0, |b(n)−Cn| < ϵn for n large enough. Hence there exists δ < 1 such that if B(z) is the generating function of b(n) and ϕ(z) = z/(1−z)^{2} (the generating function of n), we have |B(x) − Cϕ(x)| < ϵϕ(x) for 1 − δ < x < 1. Multiplying by (1 − x)^{2}, C = lim_{x→1^{−}}(1 − x)^{2}B(x). Taking logarithms

log C = lim_{x→1^{−}} ( log(1 − x) − ∑_{k=2}^{∞} log (1 − \frac{x^{k}}{k} )) = − lim_{x→1^{−}} (x + ∑_{k=2}^{∞} ( log (1 − \frac{x^{k}}{k} ) + \frac{x^{k}}{k} )).

Hence

log C = −1 − ∑_{k=2}^{∞} (\frac{1}{k} − log \frac{k}{k − 1}) = − lim_{N→∞} ∑_{k=1}^{N} (\frac{1}{k} − log N) = −γ.

Fernando Chamizo. Department of Mathematics, Universidad Autónoma de Madrid and ICMAT, 28049 Madrid, Spain.

   E-mail address: fernando.chamizo@uam.es