Why does zeta have infinitely many zeros in the critical strip? I want a simple proof that $\zeta$ has infinitely many zeros in the critical strip.
The function $$\xi(s) = \frac{1}{2} s (s-1) \pi^{\tfrac{s}{2}} \Gamma(\tfrac{s}{2})\zeta(s)$$ has exactly the non-trivial zeros of $\zeta$ as its zeros ($\Gamma$ cancels all the trivial ones out). It also satisfies the functional equation $\xi(s) = \xi(1-s)$.
If we assume it has finitely many zeros, what analysis could get a contradiction?
I found an outline for a way to do it here but I can't do the details myself: https://mathoverflow.net/questions/13647/why-does-the-riemann-zeta-function-have-non-trivial-zeros/13762#13762

why can not use a representation for $ \int_{0}^{\infty} \frac{ \xi(a+it)}{a^{2}+t^{2}}cos(xt)=g(x) $ for some g(x) and then prove this integral does not change of sign her a is different from 1/2
