Zetas and positive even integers
As a matter of fact, this method was the original Euler approach at computing the zetas of positive even integers. 7 As you can see, this ancient method requires the manipulation of the expansion of an infinite polynomial to recursively calculate the results. The computation will get very complicated as the input gets bigger. That’s why it is astonishing that Euler mana ged to use this method to compute up to 𝜁(26) 8 and later derived the general formula for 𝜁(2𝑛) that we know today:
(−1) 𝑛−1 (2𝜋) 2𝑛 2(2𝑛)!
𝜁(2𝑛) =
𝐵 2𝑛
Where 𝐵 2𝑛 are the Bernoulli Numbers. 9
What about the positive odd integers?
As mentioned above, a closed form of the zetas of any positive odd integer has not been derived yet. This is counter-intuitive having seen the zetas of positive even integers, as one would think that 𝜁(2𝑛 + 1) = 𝐴𝜋 2𝑛+1 , where A is some unknown constant. For example, in the 𝜁(3) case, which was proven by Roger Apéry in 1978 to be irrational, 10 is it possible to prove that:
𝜁(3) 𝜋 3
𝑝 𝑞
=
Where 𝑝 and 𝑞 are positive coprime integers? No one knows!
Despite little work being done on the closed form, we know these values to a high number of decimal places and have a way of expressing them using Euler numbers and Polygamma functions. 11
It is possible that the values of the zetas of positive odd integers can be encoded by a generating series. In order to do so, we have to generate an infinite polynomial that has one of its coefficients as 𝜁(3). How about:
𝐴 3 1 3
𝐴 3 2 3
𝐴 3 3 3
𝑥 3 )(1−
𝑥 3 )(1−
𝑥 3 )…
𝑓(𝑥) = (1 −
Where A is some unknown constant. I put the constant in terms of some number cubed to make the following calculation a bit easier. If we look at one of its factors:
𝐴 3 𝑛 3
𝐴 2 𝑛 2
𝐴 𝑛
𝐴 𝑛
𝑥 3 )=(1−
𝑥 2 )
(1−
𝑥)(1+
𝑥+
7 Julian Havil. (2003) Gamma Exploring Euler’s Constant , p.39. 8 Leonard Euler. (1748) Introductio . 9 What Are the Bernoulli Numbers? Accessed at https://math.osu.edu/sites/math.osu.edu/files/Bernoulli_numbers.pdf. Accessed 19/08/2024 10 Apéryodical: Roger Apéry’s Mathematical Story . Accessed at https://aperiodical.com/2016/11/aperyodical-roger- aperys-mathematical-story/. Accessed 20/08/2024 11 Special Values for the Riemann Zeta Function . Accessed at https://www.scirp.org/journal/paperinformation?paperid=109535. Accessed 20/08/2024
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