Zetas and positive even integers
Generating functions
Generating functions are functions that encode a sequence with its coefficients. 4 For example, consider the function below:
𝑥 1−𝑥 −𝑥 2
𝑓(𝑥) =
By expanding the function into an infinite series, 5 we get:
𝑓(𝑥) = 1 + 𝑥 + 2𝑥 2 +3𝑥 3 +5𝑥 4 +8𝑥 5 +13𝑥 6 +⋯
Where the coefficient of 𝑥 𝑛−1 is the 𝑛 th term in the Fibonacci sequence. The function 𝑓(𝑥)= 𝑥 1−𝑥−𝑥 2 is therefore the generating function for the Fibonacci sequence.
Finding the generating function for zetas of positive even integers
The aim is to find a function 𝑓(𝑥) , such that the coefficients of 𝑥 𝑛 encode the values of Riemann zeta function evaluated at positive even integers. The starting point is very basic but non-intuitive. Consider the Taylor expansion of 𝑠𝑖𝑛(𝑥) evaluated at 𝑥 =0 :
𝑥 3 3!
𝑥 5 5!
𝑥 7 7!
sin(𝑥) = 𝑥 −
+
−
+⋯
which converges for all value of 𝑠 by the ratio test. 6
The infinite polynomial must have roots at 0, ±𝜋, ±2𝜋, ±3𝜋, … because these are the zeros of the left- hand side. The right-hand side can be rewritten as:
sin(𝑥) = 𝐴𝑥(𝑥 + 𝜋)(𝑥 − 𝜋)(𝑥 + 2𝜋)(𝑥 − 2𝜋)(𝑥 + 3𝜋)(𝑥 − 3𝜋) …
By using difference between squares, the infinite polynomial can be expressed as:
sin(𝑥) = 𝐴𝑥(𝑥 2 −𝜋 2 )(𝑥 2 −4𝜋 2 )(𝑥 2 −9𝜋 2 )…
Divide right-hand side by −𝜋 2 ,−4𝜋 2 ,−9𝜋 2 ,… to get:
𝑥 2 𝜋 2
𝑥 2 2 2 𝜋 2
𝑥 2 3 2 𝜋 2
sin(𝑥) = 𝐴𝑥 (1 −
)(1−
)(1−
)…
Where 𝐴 is an unknown constant. To find 𝐴 , divide both sides by 𝑥 and take the limit as 𝑥 approaches 0 .
4 Discrete Mathematics: An Open Introduction. Accessed at https://discrete.openmathbooks.org/dmoi2/section- 27.html. Accessed 17/08/2024. 5 Generating Function and the Fibonacci Numbers. Accessed at https://austinrochford.com/posts/2013-11-01- generating-functions-and-fibonacci-numbers.html. Accessed 17/08/2024. 6 Maclaurin Expansion of sin(x) . Accessed at https://blogs.ubc.ca/infiniteseriesmodule/units/unit-3-power- series/taylor-series/maclaurin-expansion-of-sinx/. Accessed 18/08/2024.
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