Define $F(T)$ as the number of solutions to $\zeta(a+ ti) =0$ for $0\le t\le T$ and $0<a<1$.
How to show that $F(T)= O(T\ln T)$?
For clarity, $\zeta$ is the Riemann zeta function, $i$ is the imaginary unit and $O$ is big-O notation.
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2Perhaps you would be interested in a book, Edwards "Riemann Zeta Function" which discusses this and many interesting aspects of the RZF. – Dec 09 '13 at 01:06
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@anon sorry I was sloppy. – mick Dec 09 '13 at 01:38
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@Andrew I do not have access to books. – mick Dec 09 '13 at 01:38
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Anyway, here is a link to a nice answer about zeros (containing a further link) both of which may be of interest to you http://math.stackexchange.com/questions/442390/riemann-zeta-function-number-of-zeros/442686#442686 – Dec 09 '13 at 02:44
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@Andrew thanks for the link. ALthough Im not so familiar with the Z-function and those techniques. However that link is about the critical line alone. Whereas I am asking about the critical strip. If I got it right that is. I know there must be results about the strip too , because it is required for the PNT and other related stuff. Although it is known that more than 60 % is on the critical line ( I believe 60 % is the best known today , Hardy proved 40 % I think ) that does not seem sufficient for the error term in the PNT. See the explicit formula. – mick Dec 10 '13 at 12:10
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@Mike thanks for the edit. Looks nice now. – mick Dec 10 '13 at 12:11