# Primes from a Different World

Every textbook on number theory will begin with a treatise on prime numbers; every treatise on prime numbers will begin by emphasising their importance as building blocks or atoms of our number system: every integer can be expressed as a product of prime numbers in one way and one way only. Six is two times three and there is no other way to decompose it.1 Euclid proved this over two thousand years ago and it is so fundamental (hence the name fundamental theorem of arithmetic) to our thinking about numbers that we take it for granted. It is not!

There are numbers that behave very much like the integers but have a different structure. One rather simple example are the Gaussian integers (usually denoted $\mathbb Z[i]$) which look just like complex numbers $z=x+iy$, except that $x$ and $y$ are restricted to integer values. They live in the complex plane, but exclusively on a discrete grid amongst their continuous cousins.

1. Yes, you can write 12 as 3*4 or 2*6, but you can continue either way and eventually reach the unambiguous 2*2*3.

# From Zeta to J and Back (And Yet Again Back)

We know a lot about the $\zeta$ and $\xi$-functions, we've learnt all about the different prime counting functions, most notably $J(x)$, so it's high time we found a connection between the two. Probably not too surprisingly, the crucial link is our good friend, the Euler product

What we want to develop now is a version of this product that will suit us to find a formula that magically can count primes. (Remember that the Euler product is an analytical version of the fundamental theorem of arithmetic, so this is a natural starting point for our search.) Continue reading From Zeta to J and Back (And Yet Again Back)