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This is a video Ive been excited to make for a while. The story here braids together prime numbers, complex numbers and pi in a very pleasing trio. Quite often in modern math, especially that which flirts with the Riemann zeta function, these three seemingly unrelated objects show up in unison, and I want to give you a peek at one instance where this happens; one of the few that doesnt require too heavy a technical background. Thats not to say its easy in fact this is one of the most intricate videos Ive ever done but the culmination is worth it. What well end up with is a formula for pi, a certain alternating infinite sum. This formula is actually written on the mug Im drinking coffee from as I write this, and a fun but almost certainly apocryphal story is that the beauty of this formula is what inspired Leibniz to quit being a lawyer to instead pursue math. Whenever you see pi show up in math, theres a circle hiding somewhere, sometimes very sneakily. So the goal here is no
