# Pi

March 14 is officially Pi Day in recognition of the first three significant digits of ${\pi}$, a transcendental number whose decimal form begins $3.14159\ldots$ and goes on forever.

Definition

Pi is defined as the ratio of a circle’s circumference $C$ to its diameter $d$.

$\displaystyle\pi = \frac{C}{d}$

It’s geometrically defined. That’s it.

Appearances

But pi is an ambitious number, not content with the confines of geometry. Consider the following series that has no apparent geometric meaning:

$\displaystyle\sum_{n = 1}^\infty \frac{1}{n^2} = 1 + \frac{1}{4}+\frac{1}{9}+\frac{1}{16}+\frac{1}{25}+\cdots$

This is the sum of the reciprocals of the squares. Surprisingly, it has the curious value

$\displaystyle\sum_{n = 1}^\infty \frac{1}{n^2} =\frac{\pi^2}{6}$

Leonhard Euler discovered the above equation in 1735. Another interesting occurrence of pi is in Euler’s identity:

$e^{\pi i} + 1 = 0$

where $e$ is Euler’s number 2.71828… and $i$ the square root of negative one.

In mathematics we also have the Gaussian integral

$\displaystyle\int_{-\infty}^\infty e^{-x^2}dx = \sqrt{\pi}$

which is essentially the equation modeling the standard distribution.

Another useful occurrence of the square root of pi is in the Stirling approximation to the factorial function $x! = x(x-1)(x-2)\cdots(2)(1)$. The approximation states

$\displaystyle x! \sim \left(\frac{n}{e}\right)^n \sqrt{2\pi n}$

Pi also manages to show up in physics, largely due to the geometrical nature of our universe. For instance, the Heisenberg Uncertainty Principle says it is impossible to know both the position and momentum of a particle with high degrees of certainty. If we assign $\Delta x$ be the uncertainty in position, $\Delta p$ the uncertainty in momentum, and $h$ Plank’s constant, then we have the Heisenberg Uncertainty Principle:

$\displaystyle\Delta x \Delta p \geq \frac{h}{4\pi}$

The above is by no means a comprehensive list of usages of pi as a mathematical constant—there are countless more.

Other Pi

The capital pi ($\Pi$) is used in mathematics to denote a product, as sigma ($\Sigma$) denotes a sum. For example, we could say

$\displaystyle\prod_{k = 1}^n a = a^n$

A slightly more sophisticated product might be

$\displaystyle\prod_{k = 1}^n k = n!$

Combine this with the Stirling approximation, and we have both upper and lower case pi in one statement!

$\displaystyle\prod_{k = 1}^n k \sim \left(\frac{n}{e}\right)^n \sqrt{2\pi n}$

The lower case pi is also used as the prime counting function $\pi(x)$, which is the number of primes less than or equal to ${x}$. The explicit formula for this function is too complicated to put here. Actually, why not?

$\displaystyle\pi(x) = \sum_{n = 1}^\infty \frac{\mu(n)}{n} J(\sqrt[n]{x})$

where ${J(x)}$ is defined as

$\displaystyle J(x) = Li(x) - \sum_{\rho} Li(x^\rho) - \log 2 + \int_{x}^\infty \frac{dt}{t(t^2 - 1)\log t}$

Here, $\mu(n)$ is the Möbius function, ${Li(x)}$ is the logarithmic integral, and $\rho$ indicates the enumeration of all nontrivial zeros of the Riemann zeta function.

Hap-pi Pi Day!