Convergence of trigonometric sums over the non-trivial Zeta zeros

Question

There is video by B. Conrey where he discusses trigonometric sums of the form:

$\sum_{n=1}^\infty \cos(\gamma_n*\ln(x))$

He indicates that they have support on the prime powers: eg, 33:55 in https://www.youtube.com/watch?v=OS2V6FLFmxU

There is also a book by Mazur and Stein "Prime Numbers and the Riemann Hypothesis" (M&S) where they discuss these sums - referenced as:

$\sum_{i=1}^\infty \cos(\log(x)*\theta_i)$

The numerical evidence in M&S is compelling, but i cant see a reference to a proof that the series converge (even conditionally). They do have some end notes where they say that the phenomena is a result of the "explicit formulation", but i do not believe there is a proof of convergence.

1. is it known that these series converge?
2. if they converge in the sense of distributions, how would one prove that?
3. are there similar expressions which sum over the non-trivial zeros which do converge to something with support at the prime powers (not the sum of prime powers)?

thank you

Answer 1

The series diverges in the usual sense for every $x$,

but it converges in the sense of distributions on $(0,\infty)$, under the RH the limit is $\frac12\sum_n \Lambda(n)n^{-1/2}(\delta(\log x-\log n)+\delta(\log x+\log n))$ plus some 'trivial' terms.