here is my question. Fix any prime $p$ and consider the set of all elements of the form $\frac q{p^k}$, where $q$ is any other prime and $k$ is the unique integer such that $\frac q{p^k}$ belongs to the interval $]1,p[$. Is it true that this set is dense in $]1,p[$ and has a "nice" distribution?
Thanks in advance for any comment :)
From the Wikipedia article on prime gaps, we have that for every $\varepsilon > 0$, there is a number $N$ such that $g_n < \varepsilon p_n$ for all $n > N$ (where $g_n$ is the $n$th prime gap). This is enough to show that your set is dense.
However, it is not uniformly distributed. Look at it this way: given an integer $d$ with $1 \le d < p$, your $q/p^k$ is in $[d,d+1)$ if and only if $d$ is the first digit of the base-$p$ expansion of $q$.
Now read the article Does Benford's law apply to prime numbers? by Chris K. Caldwell at The Prime Pages web site. It discusses at length the question of whether the initial base-$10$ digits of the primes are equidistributed, and explains why there is no straightforward answer. However, it shows that if we modify the question so that it does have an unambiguous answer, the answer is: the "proportion" (in a suitable sense) of primes with initial base-$p$ digit $d$ tends to $\log_p(1+1/d)$.
