Talk:Chebyshev function

Riemann hypothesis

I doubt the equation

$\left|\sum _{\rho }{\frac {x^{\rho }}{\rho }}\right|\leq \left(\sum _{\gamma }{\frac {1}{\gamma }}\right){\sqrt {x}}=O({\sqrt {x}}\log ^{2}x)$

is meaningful. If the sum $\sum _{\gamma }{\frac {1}{\gamma }}$ is over all nontrivial roots (ordered according to the absolute value of the imaginary part, per the usual convention), then the positive and negative terms cancel, and the result is zero, which is clearly false. If the sum is intended to run only over roots with positive imaginary part, then it is divergent, as there are $\Theta (\log n)$ zeros with imaginary part in [n, n + 1]. I'm afraid that the reason for the $O({\sqrt {x}}\log ^{2}x)$ bound is much more subtle than what is alluded here. -- EJ 20:11, 30 August 2006 (UTC)