The Pi Obsession—From Circle to Riemann Zeta Function

Mathematicians’ obsession with pi starts with the most basic geometric ratio of the circumference to the diameter of a circle. For thousands of years, mathematicians have attempted to calculate its exact value via various means. Ironically, efforts approximating pi eventually yield it being among the vast ocean of irrational numbers, which are not expressible as a ratio of integers. However, early studies in algebra, starting with Diophantine Equations (polynomial equations), soon identify its uniqueness.

Early research on quadratic and cubic equations renders mathematicians realize that the set of solutions to the equations extends far beyond the real number system, demanding more numbers to be introduced. For simplicity (and laziness), Rafael Bombelli formally defines i= (-1)^{0.5} alone to establish a parallel universe of imaginary numbers by transferring notions from the real number system. Along with this new introduction, the combined sums are referred to as complex numbers, or formally considered as ordered pairs in “basic” algebra (at least Nathan Jacobson thinks so).

Considering complex numbers as ordered pairs is not Jacobson’s idea, but Leonhard Euler’s, which is based on the interchangeable nature of René Descartes’ observation of a spider web (which inspired Cartesian coordinate system— yeah, the rectangular one) and Grégoire de Saint-Vincent, Bonaventura Cavalieri, and maybe Isaac Newton’s invention of polar coordinate system. Perspective indeed plays a role, this leads to

x+yi=r(cos(q)+isin(q)).

With the assistance of some calculus, Euler proves:

e^{iπ}+1=0

This special correlation between e and pi leads to more fascinating theorems. For example, pi is a transcendental number by Lindemann-Weierstrass Theorem and zeta(2n), n∈N, Riemann zeta function with positive even number input, being associated with pi^2n.

zeta(2)=1+1/(2^2)+1/(3^2)+…+1/(n^2)+…