# How to Impress Your Friends

Want to show your friends a thing or two? Try solving one of the Millennium Prize problems: the Riemann hypothesis! Your solution will be worth \$1 million of money, so it’s kind of a big deal.

Step 1: The Riemann zeta function is the function of the complex variable s, defined in the half-plane R(s) > 1 by the series:

$\zeta(s) = \displaystyle\sum_{n=1}^{\infty} n^{-s}$

and in the whole complex plane by analytic continuation. Copy this onto A4 paper in a clear hand. You will notice that the Riemann zeta function converges to zero where s is an even negative integer. These are the trivial zeroes.

Step 2: Prove that all non-trivial zeroes of the Riemann zeta function have real part equal to ½.

Hint: If you get stuck on Step 2, try disproving the Riemann hypothesis instead. A single counterexample will do the trick, and you still get the money!

Step 3: Once you’ve proved (or disproved) the Riemann hypothesis, ask your friends whether they’re impressed. If they say no, use the prize money to buy a car or a hot spouse or whatever shit they think is cool.