In a previous post we discussed some methods for showing that

where the **zeta function** is defined to be

for .

Apéry’s constant is the value of the zeta function at

named for Roger Apéry who proved this number irrational in 1978 [1].

Prior to Apéry’s result, it was known that the zeta function, evaluated at the positive even numbers, was irrational with closed form

for where are (rational) Bernoulli numbers (see [3] for proof).

For example , , , etc.

Yet no simple form is known for Apéry’s constant or indeed any of the other odd zeta values, with the exception of . Indeed, it has been postulated that no such simple form is even possible. Nevertheless, there are various closed form solutions involving infinite sums or integrals.

So why is finding a simple expression for the odd values of the zeta function so much more difficult than the even case? We do not know the answer to this question, but what we can do is try to generalise the proofs previously given for and see where they fail. This is our focus for the remainder of this post.

Consider the proof that based on Fourier series given here for example. To construct an equivalent statement for it seems natural to consider the Fourier series of over the domain so that

is convergent to on where

for and

for Thus for .

Integrating by parts thrice we have (up to an additive constant)

And therefore

Thus we have the Fourier series expansion

We see now a clear difference in the situations, as we have inherited a sum over (the harmonic series) which would seem to complicate the analysis. Considering is no longer effective as whenever . Perhaps then we might take so that

We know that so then

or equivalently

Thus, in spite of our original goal, we are lead to a well-known value for the Dirichlet beta function.

It is natural then to enquire about the value of the sum

This sum is known as Catalan’s constant and it is unknown whether it is irrational.

The even values of the Dirichlet eta function are known, for example

It is interesting to take in the Fourier series given above, as we know , and so

where cycles through the values so that

Wolfram Alpha gives

and a monstrosity for

Next, we consider the Fourier series expansion of , in which case

so that

Thus for and

Then evaluating the Fourier series for at gives

And thus

up to the sign error… that’s not bad.

**References**

[1] Apéry, R. “Irrationalité de ζ(2) et ζ(3)”, Astérisque, 61: 11–13.

[2] Euler, L. “Exercitationes Analyticae” http://eulerarchive.maa.org//docs/originals/E432.pdf

[3] van der Poorten, A. “Apéry’s proof of the irrationality of ” https://web.archive.org/web/20110706114957/http://www.maths.mq.edu.au/~alf/45.pdf