Periodicity in Riemann zeros.

Question

Has someone studied if the non-trivial zeroes of the Riemann zeta function has some "periodicity" or "quasiperiodicity"? And what about generalized zeta functions and/or L-functions?

Answer 1

Sure, there is no periodicity or quasi-periodicity, see the density of zeros which is $2\pi T\log T$. The structure is similar for all the L-functions.

The fact that the discrete distribution encoding the non-trivial zeros $$2\pi\sum_t \delta(\omega-t)$$ is the Fourier transform of the (almost) discrete distribution $$\frac12 e^{|u|/2}-f(u)-\sum_n \Lambda(n)n^{-1/2} (\delta(u-\ln n)+\delta(u+\ln n))$$ (where $f(u)= (e^{u/2}\log(1-e^{-2u})'$ is a fast decaying distribution)

is not well-understood, it does give for sure some kind of repealing structure to the zeros.