One of the weirdest things in mathematics is how completely unrelated fields tend to connect with one another.
A particularly interesting case is the play between number theory (the study of natural numbers N={0,1,2…}) and complex analysis (the study of functions on C={a+bi:a,b∈R}). One is discrete and uses modular arithmetic and combinatorics; one is continuous and uses integrals and epsilon-delta proofs. And yet, many papers in modern number theory use complex analysis. Even if not directly, basically all of them rely on some result with a complex analytic flavor.
What's going on?
This post gives an example of a fundamental result in number theory that uses complex analysis, and tries to explain where that usage is coming from. My writing is semi-technical, I'm not trying to formalize everything, but mostly doing this as an exercise to clear out my own confusion (and hopefully to convince others of why this mathematical connection is valuable).
Technical details in the proof follow Ang Li’s notes; exposition, intuition and possible mistakes are mine.
I Dirichlet's Theorem
We know that there are infinitely many prime numbers since around 300 BC. Over 2000 years later, Dirichlet proved that there are infinitely many primes even if we restrict ourselves to an arithmetic progression of the form a,a+q,a+2q,…, as long as a,q have no common factors.
I'll focus on a concrete example: there are infinitely many prime numbers ending in the digit 7, that is, primes of the form p=10n+7.
This statement can be understood by a curious 7-year-old: she could learn about prime numbers, then write out examples of them - her own age, 17, 37, ... and then be told that, although such numbers ending in 7 become harder and harder to find, they will always be there, beyond whatever limits we imagine.
If the statement is so easy to phrase, why does it require complex analysis?
Do we really need to use a number defined as the square root of -1, as well as computing infinite sums and