Apéry's Constant

Today I turned 25355 days old and this number turns up in the Engel expansion of \( \zeta(3) \). Firstly however, let's remind ourselves that \( \zeta \) is the Riemann zeta function and can be expressed as:$$ \zeta(3)=\sum_1^{\infty} \frac{1}{n^3} $$ $$ \text{or  } \zeta(3)=\lim_{n \rightarrow \infty} \left( \frac{1}{1^3}+\frac{1}{2^3}+ \cdots + \frac{1}{n^3} \right) $$This works out to around 1.202056903159594285399738161511449990764986292 and is known as Apèry's constant. I've written about this constant before in a post titled The Basel Problem and Beyond on May 7th 2017. According to Wikipedia:

This constant arises naturally in a number of physical problems, including in the second-and third-order terms of the electron's gyromagnetic ratio using quantum electrodynamics. It also arises in the analysis of random minimum spanning trees and in conjunction with the gamma function when solving certain integrals involving exponential functions in a quotient which appear occasionally in physics, for instance when evaluating the two-dimensional case of the Debye model and the Stefan–Boltzmann law.

This doesn't mean much to me but there you have it. As for the Engel expansion, I made a blog post about it in 2016. The expansion consists of the terms that go to make up the denominators of the fractions that when added will approximate \( \zeta(3) \). OEIS A053980 lists the first few terms as: 1, 5, 98, 127, 923, 5474, 16490, 25355

A053980

Engel expansion of zeta(3) = 1.20206...

This means that \( \zeta(3) \) can be approximated as:$$ \zeta(3) \approx \frac{1}{1}+\frac{1}{1 \times 5} +\frac{1}{1 \times 5 \times 98}+\frac{1}{1 \times 5 \times 98 \times 127} +\frac{1}{1 \times 5 \times 98 \times 127} + \cdots $$Of course, once the zeta function is touched upon, one can find oneself in very deep water very quickly so I'm not going to say too much more except to include a screenshot (Figure 1) from Wolfram MathWorld showing the Engel expansions for some of the other constants:

Figure 1

The SAGE code for generating these sequences is fairly straightforward. Figure 2 shows what's involved for Apèry's constant:

Figure 2: permalink

Of course, simply replacing u in the above code by say \(e\) or \(\pi \) will produce the associated Engel expansion.

on May 12th 2021