Can someone help me understand and evaluate the Riemann zeta function?

[epkid08]

I still don't understand a few things.


Let's say we had a non-trivial zero counting function, $$Z_n(n)$$, for the riemann zeta function. Couldn't we fairly easily prove the riemann hypothesis by evaluating $$\zeta (\sigma+iZ_n)$$, solving for $$\sigma$$, then proving it for all n using induction?


On another note, I still need help in evaluating the actual function. Can someone show me, step by step, how to evaluate say, $$\zeta (1/2 + 5i)$$? Please be specific

 

[yasiru89]

You can use (21) or (25) on http://mathworld.wolfram.com/RiemannZetaFunction.html in the evaluations, though you'll have to resort to numerical methods sooner or later.

Of the attempt at proving the Riemann hypothesis, I can only say that the approach you suggest is similar to reducing the problem to Merten's conjecture (by a Mobius reciprocation)- a proof of which would imply the Riemann hypothesis! However, Merten's conjecture has been shown false (though its falsity does not imply the falsity of the Riemann hypothesis).