Using generating functions to calculate Zetas of positive even integers
Zezi Li
The Riemann Zeta function is arguably one of the most important functions in analytic number theory. It started off as a generalization of the harmonic series and Basel problem across all positive integers. To this day, the exact closed form of zetas of positive odd integers (except 1 which makes the function diverge) are still unknown, while the zetas of positive even integers are solved, and a closed form can be computed. Mathematicians then extended its domain to all real numbers and in the complex plane, where they developed a new definition for the function using analytic continuation to include all complex numbers. 1 The Riemann Zeta function can be used to calculate 𝜋 to an infinite degree of accuracy. It also has an entire hypothesis tied to it, known as the Riemann Hypothesis, which if solved can prove many other proofs that assume the Riemann Hypothesis is correct. However, it is calculating the closed forms of the zetas of positive even integers I want to focus on.
The Riemann Zeta function evaluated at positive even integers
The Riemann Zeta function of 𝑠 is defined as the infinite sum of every positive integer to the power of negative s, as given below: 2
𝜁(𝑠) = ∑ 𝑛 −𝑠 ∞ 𝑛=1
Consider when 𝑠 is a positive even integer, we have: 3
𝜋 2 6 𝜋 4 90
𝜁(2) =
𝜁(4)=
𝜋 6 945
𝜁(6)=
… Looking at these results for the first time, one must be surprised at how the circle constant 𝜋 comes into play, and cannot help but ask: does this pattern of 𝜋 2𝑘 being part of the value continue? Is there a general formula? To most mathematicians’ delight, the answers to both questions were found to be ‘ yes ’ . And there are multiple ways to compute the formula. One of these is a generating function.
1 But what is the Riemann zeta function? Visualizing analytic continuation. Accessed at https://www.youtube.com/watch?v=sD0NjbwqlYw. Accessed 15/08/2024.
2 Riemann Zeta Function. Accessed at https://brilliant.org/wiki/riemann-zeta-function/. Accessed 15/08/2024. 3 Computing zeta at even numbers. Accessed at https://www.johndcook.com/blog/2022/06/24/zeta-at-even- numbers/. Accessed 16/08/2024.
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