# Why should I switch to Python?

Gareth McCaughan Gareth.McCaughan at pobox.com
Wed Apr 5 15:05:44 EDT 2000

```Michael Hudson wrote:

> It's also possible to show that if a and n have no common factor, then
> there are an infinte number of primes in the artihmetic progression
> a+dn as d=0,1,2,3,4,..., but the proof is, umm, somewhat harder (ie. I
> don't know it; I think it involves slinging Riemann zeta functions).

Dirichlet L-series, actually. Closely related to the Riemann
zeta function, but not quite the same thing. You look at
sum{1..oo} of a(k)/k^s where a is a "character mod n" (which
just means that a(k) depends only on k mod n and a(pq) = a(p)a(q)),
and the crux of the proof is showing that this thing is well
behaved at s=1 if a isn't the obvious thing that's always
equal to 1. Once you've got that the rest of the proof isn't
much harder than the following witty proof that there are
infinitely many primes: "sum{1..oo} of n^-s = product{primes}
of 1/(1-p^-s). As s->1, the former goes to infinity, so the
latter does too, therefore there must be infinitely many
factors in the product. QED."

Therefore you should switch to Python. QED. :-)

--
Gareth McCaughan  Gareth.McCaughan at pobox.com
sig under construction

```