With the amount of math Euler created, he must’ve pulled a proportionately large number of late nights; I think it’s safe to say he would’ve been quite the Taco Bell conniseur. But that only begs the question…what would his go-to order have been? That’s a tough one.

Anyway, I was at Taco Bell with a good friend of mine when we got to talking about the beauty of the Euler product formula and its proof. It reads:

$\zeta(s) = \sum_{n=1}^\infty\frac{1}{n^s} = \prod\frac{1}{1-p^{-s}}, \forall p \in \mathbb{P}$

An infinite sum over the naturals ends up equalling a product over the primes…it’s nerdy, but I get an adrenaline rush when thinking about that. And it’s actually quite simple. I won’t go into it much, as the Wikipedia article linked above does a great job at illustrating the proof. In short though, Euler takes the first element of the infinite series (excluding 1) and multiplies a copy of of the series by that element. He then subtracts the copy from the original, yielding a new series sieved of all multiples of that element. Rinse and repeat this algorithm to generate the primes!

Then, the computer scientists in us revealed themselves:

• You know what would be more interesting: how many times would a number that’s already been exluded have been hit?
• What do you think the runtime of that is?

In other words, mapping the natural numbers $$\to$$ lists of their factors. After arguing over the superiority of the natural vs. binary logarithm, our intuitions told us the runtime was $$O(n\log{n})$$; however, we never proved that rigorously. Leave a comment if you’ve got one!

Logarithm: the number of times you must divide a number by a base, until that number goes to 1

For example: $$\log_2{8} = 3$$ is like $$8/2/2/2 = 1$$

Eventually I found myself researching complex logarithms. For a complex number $$z = re^{i\theta}$$, there are infinitely many outputs of $$\ln{z}$$ that all differ by integer multiples of $$2\pi i$$, which gives the plot below a sort of “height”. In the complex domian, the logarithm looks much weirder than it does over the reals - it resembles a spiral staircase, like that one in Super Mario 64, except without the steps:

It was fun researching and learning about the complex logarithm; to know there is so much more hiding within such an elementary function is exciting. What else could be hiding beneath the surface?