Primes, primes

A prime number is a positive integer p that is divisible only by \pm p and \pm 1.
Now multiplication and addition interact rather strangely, in almost unfathomable ways. As a consequence, the distribution of prime numbers among the integers is somewhat random, and mathematicians have tried to find patterns in this distribution for a very long time. The starting point is a fact that the Greeks of classical times already knew: there are an infinite number of primes. Euclid (c. 300 BCE!) includes a proof of this (it is Proposition 20) in Book IX of the Elements.

Even now, there remain many puzzles concerning primes about whose solutions we seem to know next to nothing.

Surprisingly, even though prime numbers are very discrete objects, some of the major tools in investigating them have been certain continuous functions, and in particular the zeta function
\zeta(s) first introduced around 1750 by the Swiss mathematician Leonhard Euler. It is probably the most famous function in all of mathematics, at least among mathematicians, but undoubtedly more mysterious than not to the rest of the world. … (AMS)