It is known that cos(N) spans a countable dense set in [-1,1].
(N: any natural number)


As far as I know generally, for any continuous function f defined in [a,b],


f is Riemann integrable where its domain is a countable dense set in [a,b].


**My question: will cos[t\_n\*Log(p)] Spans a countable dense set in [-1,1]?**
\**(Variable: n; 1 to infinity)*\*


t\_n=Im[Zetazero(n)]: the imaginary part of the n'th nontrivial zero of the Riemann zeta function.
p: any prime number



Cos[Im[zetazero(n)]Log(prime)] spans a countable dense set in [-1,1]?
For any fixed real $\alpha$, the fractional parts of the numbers $\alpha \gamma$, where $\beta+i\gamma$ runs over all zeros of $\zeta(s)$ in the critical strip with $0<\gamma < T$, become uniformly distributed in $\mathbf{R}/\mathbf{Z}$ as $T\to \infty$. This is a theorem of Fujii; see his paper "On the zeros of Dirichlet L-functions, III", Transactions of the AMS, vol. 219. This affirmatively answers your question.


