Most of us are familiar with from school days. You may remember it as defining the area of a circle (), and you may recall its value: 3.14159265359… with seemingly no end. looms large in many branches of science, from astronomy to particle physics, to engineering and finance. It belongs to the league of special numbers that occupies a hallowed place in mathematics (others in this league include the exponential constant (2.71828 …, and the golden ratio: 1.6180339 …)

But most people have no clue where this supreme number comes from and whether it is related to something deeper. This is the motivation for today’s post.

Cutting to the chase, here’s a fundamental equation for :

In words, this equation says is equal to the square root of 6 times the (Reimann) zeta-function with exponent 2. Bernhard Reimann (1826 – 1866) was a German mathematician who made fundamental contributions to analysis, number theory, and differential geometry. His pioneering results on differential geometry laid the foundations of the mathematics of Einstein’s theory of general relativity.

In general, the Reimann zeta function with exponent x (where x is greater than one) is:

where n runs to infinity. Thus, the zeta function may be written algebraically as:

Euler’s formula for may be considered a special case of the zeta function, where the exponent is 2.

We owe it to the brilliant German mathematician, Leonhard Euler (1707 – 1783) for this equation which is known as Euler’s product formula. While toying around with the zeta function, Euler noticed that there is something odd about it. First, it veers off to infinity when x = 1 in which case, the zeta function becomes the harmonic series, made famous by the ancient Greek mathematician, Pythagoras who discovered that this infinite sum has a direct correspondence with music (hence the expression ‘the music of the spheres’). The zeta function also adds up to infinity when x < 1. For example, when x = -1, the zeta function becomes the series 1 + 2 + 3 + 4 + … which clearly sums to infinity. You can verify this result for other cases where x < 1 on a computer. The only time when the function is “well-behaved” (gives a finite value) is when x > 1 as in Euler’s formula for .

Euler’s next brilliant insight is to show that the zeta function is deeply connected with the behaviour of prime numbers. Prime numbers, to briefly recall, are numbers like 2, 3, 5, etc. which are divisible only by 1 and itself. That is, a prime number is any whole number greater than 1 whose only factors are 1 and itself. These numbers (there are thought to be an infinite number of them) are the basic building blocks of Number Theory.

So, what is the big deal about the connection between latex \pi is essentially a chaotic number or more technically, an irrational number because its decimals trail to infinity. Yet, embedded in the zeta function is an orderly progression of primes. To be more precise, Euler proved that the inverse of the zeta function is equal to the actually the probability that two randomly chosen integers have no prime factors in common.

Denoting this probability by P, Euler showed that:

That is, P can be expressed as an infinite product over all primes (this is known as the Basel problem). Euler then proved that 1/P is none other than the zeta function to the power of 2!

But we already know that 6 times the zeta function with exponent 2 is equal to 6 x 1/P. Inverting this result gives

which is the surprising connection between and primes. Since prime factorization is of immense practical important (for cybersecurity for example), Euler’s results add another feather to the cap of the magical number that is .

Further Study

[1] For a proof of the Basel problem, check out this blog: https://mattbaker.blog/2016/03/14/probability-primes-and-pi/

[2] For a proof of Euler’s product formula for the zeta function, see the Wikipedia entry “Proof of the Euler product formula for the Riemann zeta function” at https://en.wikipedia.org/wiki/Proof_of_the_Euler_product_formula_for_the_Riemann_zeta_function