Let $N(\sigma_{1}, \sigma_{2}; T)$ denote the number of nontrivial zeros of the Riemann zeta-function in the rectangle $1/2 \leq \sigma_{1} \leq \sigma_{2} < 1, 0 < Im(s) < T$.

If one could show that $N(\sigma_{1}, 1; T) \ll T$ for each $\sigma_{1} > 1/2$, then would it follow that $N(1/2, 1/2; T) \sim (1/2\pi)T \log T$?

($N(1/2, 1/2; T)$ denotes the number of zeros on the segment $\sigma = 1/2$, 0 < Im(s) < T.)

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    $\begingroup$ Not quite, for silly reasons as literally given: the constant needn't be uniform, etc. $\endgroup$ – paul garrett Jul 14 '16 at 23:07
  • $\begingroup$ @paulgarrett I agree with your scepticism, but "follow" may be somehow different from "follow immediately". $\endgroup$ – Fedor Petrov Jul 14 '16 at 23:28
  • $\begingroup$ @FedorPetrov : I guess an illustration is $\rho_{k} = 1/2 + e^{-k} + i\frac{ k \ln k}{2 \pi}$, it fulfills the requirements, everything is uniform, but the RH is 100% false. I saw those kind of sequences in zeros of entire functions represented by Fourier transforms, Peter Hallum trying to understand if functions close to $\xi(s)$ have their zeros on the critical line $\endgroup$ – reuns Jul 15 '16 at 23:04

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