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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

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I guess you mean "the imaginary part of the nth nontrivial zero of the Riemann zeta function on the critical line"? Cause RH has not been proven yet... – Sylvain JULIEN Apr 17 '11 at 14:23
@Seongsoo, just to clarify, have I got this right: you want to fix a prime $p$, then look at the set of numbers $\cos(t_n\log p)$ as $t_n$ runs through the imaginary parts of the non-trivial zeros of zeta, and you want to know whether this set is dense in $[-1,1]$. If that's right, then my question for you is, why? Is there something that makes these numbers especially interesting? – Gerry Myerson Apr 17 '11 at 23:30
Yes. That's exactly what I meant. I recently found that if this is right, I think it is possible for us to apply integration methods to the absolute value of the Riemann zeta function of which domain is restricted at the set of its nontrivial zeta zeros. – Seongsoo Choi Apr 18 '11 at 7:48
up vote 4 down vote accepted

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.

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I think the theorem you mention is originally due to Hlawka in 1975. It was observed by Rademacher, in the 1950s, that this follows from the Riemann Hypothesis. See the intro to this paper: – Micah Milinovich Apr 19 '11 at 2:18
Hlawka also assumes RH (his interest being in proving a certain rate of equidistribution); Fujii's proof is unconditional. In the sequel (joint with Sound) to the paper you reference, you'll find a reference to the Fujii paper I cited. – David Hansen Apr 19 '11 at 2:34
In the sequel Ford, Sound, and Zaharescu attribute the theorem that $\{\gamma \alpha\}$ is uniformly distributed (mod 1) to Hlawka. Hlawka has another result on discrepancy of this sequence which relies on RH which was later proved unconditionally by Fujii. – Micah Milinovich Apr 19 '11 at 2:53
Whoops! Yes, you are right. Examining the MR of Hlawka's paper, it seems Hlawka and Fujii proved essentially the same result at essentially the same time. – David Hansen Apr 19 '11 at 3:15
Thanks for the references, this is exactly the result that my Google search seemed to indicate but without any concrete references. Is it safe to assume that the methods used by Fujii et al. in these references are also sufficient to show that one also has uniform distribution in $\mathbb{R}/k\mathbb{Z}$ for any real $k$ (since for the OP's question it seems that one needs uniform distribution in $\mathbb{R}/2\pi\mathbb{Z}$ instead of $\mathbb{R}/\mathbb{Z}$)? – ARupinski Apr 19 '11 at 5:46

If the cosine values only related to the zeros on the critical line span a countable dense set, every cosine value determined by all the nontrivial zeros can span a countable dense set, too. n can be varied from 1 to infinity, and this I think will make a dense set. Do you agree?

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Actually a stronger statement holds: $\cos(t_n\cdot\log(p))$ is dense in $[-1,1]$ for $n$ fixed. The idea behind proving this stronger version is to show that for any $x\in[-1,1]$ and $\epsilon>0$ fixed, there is a prime $p$ such that for some $k$ one has:

$(1)\;\;\;\;\;\;\;\;\;\;\;\;\;|t_n\cdot\log(p)-\arccos(x)-2\pi k|<\epsilon$

The Prime Number Theorem implies $\log(p_{m+1})-\log(p_m)< \frac{1}{2}t_n^{-1}\cdot\epsilon$ if $m$ is large enough (where $p_m$ is the $m^{th}$ prime). Hence for some large enough $k$ one can satisfy $(1)$ with $p$ prime. Since the stronger statement holds, your original statment holds as well.

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I thought OP meant $p$ to be fixed and $n$ to vary. – Gerry Myerson Apr 17 '11 at 23:25
I guess I read it as $n$ and $p$ both varying, but if you are right Gerry then I don't know anything about the distribution of the zeros and so can't really give any insight. I would assume it is still true, and I would venture to guess that someone else probably knows enough about the zeros of the zeta function to be able to prove it roughly the same way. – ARupinski Apr 18 '11 at 0:07
As an update to my previous comment, after searching Google, I have found some statements that seem to indicate the zeros of the zeta function are fairly nicely distributed so that if one fixes $p$ (as per Gerry's interpretation) and only varies $n$, then $\cos(t_n\cdot\log(p))$ is dense in [-1,1] by a modification of my basic argument. Unfortunately none of these comments has any references for this result, so I cannot guarantee it is definitely true. – ARupinski Apr 18 '11 at 3:55
According to my computational results from Mathematica, it really seems that such sets are dense in [-1,1]. However, I actually don't know well about the behavior of the nontrivial zeros, so I have difficulty proving this denseness, currently. I guess it may be easy for someone who has studied about the zeros for a long time to prove this. – Seongsoo Choi Apr 18 '11 at 8:06
David's answer and its comments gives the necessary references for the necessary uniform distribution of the zeros. – ARupinski Apr 19 '11 at 5:47

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