In a previous post we discussed some methods for showing that
where the zeta function is defined to be
Apéry’s constant is the value of the zeta function at
named for Roger Apéry who proved this number irrational in 1978 .
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  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.
is convergent to on where
for Thus for .
Integrating by parts thrice we have (up to an additive constant)
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
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
Thus for and
Then evaluating the Fourier series for at gives
up to the sign error… that’s not bad.
 Apéry, R. “Irrationalité de ζ(2) et ζ(3)”, Astérisque, 61: 11–13.
 Euler, L. “Exercitationes Analyticae” http://eulerarchive.maa.org//docs/originals/E432.pdf
 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