Suppose you worked at a bank that only had quarters and dimes. There's lots of amounts of money you can pay out: You can make payments of a dollar, or $1.50, or $2.60, or 45¢, lots of things.
And you can combine payments too: Since you can pay $1.50 and you can pay $2.60, you can also combine them into a payment of $4.10. Put another way, we can decompose a payment of $4.10 into two separate payments of $1.50 and $2.60.
3 * 4 = 2 * 6. You can multiply different bunches of numbers and get the same result. But you can't do this if the numbers you multiply are all prime. Why is that?
A basic fact about numbers, yet almost nobody graduates from 12+ years of compulsory math education able to explain it. Math class is a waste of time, fixated on making you a human calculator, John Henry against the electronics industry, memorizing tables and running algorithms.
Math class is boring and everyone is right to be bored by it. You have a brain to do better things with.
Everyone talks about the Riemann zeta function like it's tricky to construct and requires thinking about the sophisticated machinery of meromorphic continuation. It's not and it doesn't. It's much simpler than that.
Let ζ(-p) = 1^p + 2^p + 3^p + … be the Riemann zeta function. This converges when the real component of p is < -1. How do we make sense of this for general powers p?
Well, let's consider a more general problem. Things are often simpler in greater abstraction, at least to my mindset. Suppose you want to make sense of f(x) + f(x + 1) + f(x + 2) + … in general, even though this may not converge. What can you do?