Is there a trigonometric identity that implies the Riemann Hypothesis?

Question

I was reading the Wikipedia article on the Prime Number Theorem and I saw this. They use a trigonometric identity to prove that the Zeta function has no zeroes on the line Re(s)=1. Why isn't there a similar identity that shows that ζ(s)≠0 when Re(s)=0.5+ε (where ε is positive)?

Answer 1

The referenced article uses the fundamental theorem of arithmetic, or part of it, not the prime number theorem. ..

I would just note that there is lots of complex analysis at work there, not just trigonometry. ..

Also, your question is kind of open ended... sort of like, "why is the sky blue?"

No one has been able to prove it (RH) yet...

Answer 2

The trigonometric identity is only one (though admittedly very clever) ingredient of that proof. The main idea is to use the Euler product for $\zeta(s)$, and that's convergent only for $Re(s)>1$.