How many digits of Pi do we need?
I don’t know \and neither do you.
Recently I came across this NASA written article answering a question about how many digits of Pi do we need. The answer, this being NASA and all, was all about the circumference of the universe and concludes that we need about 39 digits of Pi. I’ve never seen an argument so myopic coming from a scientific source. I love NASA to bits, but whoever thought of this article doesn’t know maths.
Yes, Pi is defined as the ratio between the diameter of a circle and its circumference, but to think that’s all that Pi is is not knowing math and if you don’t know maths, you shouldn’t write about Pi, I don’t think. Let me tell you about someone that knows maths.
The name is Bernhard Riemann and from Gauss he got a curiosity about the distributions of prime numbers. He discovered a very nice relation between the distribution of prime numbers and the (non trivial) zeroes of a function that we now call the Riemann Zeta function. Provided such zeroes all fall in the line of complex numbers that have real part equal to 1/2. That’s an open problem in Math called the Riemann Hypothesis. We still don’t know the answer to that. What do you say? You don’t know what complex numbers are? That because you don’t know maths.
The Riemann Zeta function is defined as the analytic continuation of the function defined (where it makes sense) as
“But wait,” you say… “Weren’t you going to talk about the digits of Pi?” Yes I am! Be patient!
You see, another guy that knew maths, Leonard Euler, demonstrated that the value of the Zeta function at 2 is exactly Pi squared divided by six. There are several ways to prove this with elementary calculus, or Fourier Series, or Complex Analysis. This was known as the Basel Problem. What could Pi have to do with prime numbers? I don’t know, you don’t know, even people that know a lot about maths do not know!
So, you see, it may be, though I don’t have even an inkle of proof that it is so, that the calculation of the (non trivial) zeroes of the Zeta function have something to do with the digits of Pi. I’m not saying this is sure, I’m saying nobody knows if it could be. So, how many digits of Pi do we need? We don’t know, because Pi appears in a lot of problems that may, or may not, be solved in a future where a third grader knows more maths than I do.
