Alas, the REAL \pi day of the century – the only one that most of us currently living will ever see – is over.
But don’t despair! There are other \pi’s, in fact infinitely many, and they have been tragically ignored in favor of the so-called REAL \pi. These other \pi’s happen to be formal power series, so there should be a number of perfectly legitimate FORMAL POWER SERIES \pi days of the century, and we can’t possibly have missed them all, right?
Since at least the 1930s, people have been interested in formal power series analogues of real numbers. A real number like \pi is an infinite sum of powers of 10, each multiplied by a coefficient:
If we replace 10 with x, then we get a formal power series. We choose a prime p, and the allowable coefficients are 0, 1, \dots, p - 1. The difference is that when we add or multiply power series, we use arithmetic modulo p for the coefficients.
Well, in this “number system” of formal power series modulo p, there is a “number” that plays an analogous role to the real number \pi. It’s defined as an infinite product. Here are the first few terms in the expansion of the p = 2 analogue, \pi_2:
In[1]:= With[{p = 2}, MapAt[Mod[#, p] &, Series[Product[1 - (x^p^n - x)/(x^p^(n + 1) - x), {n, 1, 5}], {x, Infinity, 20}], 3]]
%[[3]]
In a 1935 paper, Leonard Carlitz defined formal power series analogues of the exponential function, the logarithm, and even the Riemann zeta function. The remarkable fact is that the special power series \pi_p is related to those functions in ways that parallel the familiar properties of their counterparts, for example the exponential function being periodic with period 2 \pi i, and the fact that the Riemann zeta function evaluated at 2 m is a rational number times \pi^{2m}. It’s also known that \pi_p is transcendental, just like \pi, but over the field of rational power series rather than the field of rational numbers.
So there we go! The \pi_2 day of the century was 1/01/11 – January 1st, 2011. Huh, I guess we already missed that one…
What about \pi_p for other primes p?
In[3]:= With[{p = 3}, MapAt[Mod[#, p] &, Series[Product[1 - (x^p^n - x)/(x^p^(n + 1) - x), {n, 1, 5}], {x, Infinity, 20}], 3]]
%[[3]]
In[5]:= With[{p = 5}, MapAt[Mod[#, p] &, Series[Product[1 - (x^p^n - x)/(x^p^(n + 1) - x), {n, 1, 5}], {x, Infinity, 20}], 3]]
%[[3]]
Nope; the nonzero terms are too sparse to extract a meaningful date! Unfortunately, this problem will probably just get worse as we increase p. So maybe we need to look elsewhere for analogues of \pi that contain good dates in their expansions. But on the bright side, you only have to wait 96 years instead of a full century for the next big \pi day.
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