# Does $N_0(T) =N(T)$ where $N$ is the exact Riemann $\zeta$ zero counting function and $N_0$ is the approximate zero counting function imply the RH?

The main new result is that the zeros on the critical line are in one-to- one correspondence with the zeros of the cosine function, and this leads to a transcendental equation for the n-th zero on the critical line that depends only on $n$. If there is a unique solution to this equation for every n, then if $N_0(T)$ is the number of zeros on the critical line, then $N_0(T) =N(T)$, i.e. all zeros are on the critical line.

Has this criteria been accepted as proving the RH if it were proven to be true?

Last I checked with the author, he said people in the math community still haven't accepted even that criteria and it seems pretty clear to me that it is a good criteria.

• (126) in the paper seems to come out of nowhere. Where is it proven? – punctured dusk Jul 31 '18 at 17:52
• @barto: The argument follows in the next few pages. However, the overall feel of the paper is "Here's an equation that all zeros on the critical line must satisfy. Notice that things go bad in this equation if nontrivial zeros exist, so they probably don't." – Alex R. Jul 31 '18 at 17:58
• @AlexR. As I understand it, everything that follows is under the assumption that they're working with a zero satisfying (126). Given previous comments I give up trying to understand :p – punctured dusk Jul 31 '18 at 18:10