Question

I have Edwards and Titmarsch books on Riemann zeta function with me. I could not find (maybe I did not read through that carefully), but are there results similar to the form like the one given below:

Is there a non-trivial zero $\rho$ starting from which $(\Im(\rho)\log(2)/2\pi )$ runs only through integer values

I would love any elaboration or links on this topic.

Thanks,
Roupam Ghosh

Answer 1

Such a conjecture is false.

EDIT: A simpler argument - a more precise asymptotic for the number of zeroes $N(t)$ of imaginary part $\le t$ (counted with multiplicity) is $$N(t) = \frac{t}{ \pi} \log \frac{t}{2 \pi e} + o(\log t),$$ This is enough to show that, for any fixed $\epsilon > 0$, $$N(t + \epsilon) - N(t) \sim \frac{\epsilon}{\pi} \log t,$$ and thus, for sufficiently large $t$, and for any interval of length $\epsilon$, there are zeroes (whose imaginary part lies) in this interval, which also implies the conjecture is false. THIS DOES NOT USE GRH.

PS: Scott Carnahan helpfully remarks that the wikipedia article points out that Littlewood noticed that the difference in the imaginary parts of the zeros tends to zero as $t \rightarrow \infty$ (presumably by exactly using this asymptotic result of von Mangoldt above). Personally I prefer mathematics rather than an appeal to authority, but apparently that is not enough for some.

REMARK: Dear Wadim, please read this again, and realize that it DOESN'T USE GRH. The estimate of zeros (which was basically known by Riemann) is about zeroes in the CRITICAL STRIP (real part in $[0,1]$) not the CRITICAL LINE. Having done this, you can delete all your comments, I'll edit this answer, and we can all pretend it never happened. (In fact, I'll make this community wiki so you can delete this remark yourself.)