Question

Prove that the nontrivial zeros of the Riemann zeta function have a real part of 1/2.

Answer 1

Well, for all complex numbers \(s=\sigma+it\) with \(\mathfrak{Re}(s)=\sigma>1\), we have the absolutely convergent well known formula:\[\large\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^s}=\left(\sum_{n=1}^{\infty}\frac{\cos(t\,\ln n)}{n^{\sigma}}\right)-i\left(\sum_{n=1}^{\infty}\frac{\sin(t\,\ln n)}{n^{\sigma}}\right).\]

Answer 2

From which the proof easily follows.

Answer 3

I was kidding... There is no proof of the Riemann hypothesis........
http://en.wikipedia.org/wiki/Riemann_hypothesis

Answer 4

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Answer 5

If anyone ever solves it, there's a million dollar reward. The problem was first formulated in 1859, and no one has solved it since. All we know is that all the zeroes are either negative even integers, or have a real part greater than zero and less than one.
http://planetmath.org/encyclopedia/CriticalStrip.html