Ratios of primes

[Chris Long]

In article <1990Apr19.0...@Neon.Stanford.EDU>, Ilan Vardi writes:

> The answer is yes, p_i/p_j is dense in R+. I think that this is
> a special case of the following lemma with f(x) = x log x:

Another way of looking at it is to note that p(x) ~ x*ln(x),
and so lim p([rx])/p(x) = r. If I'm not mistaken, the more general
result you mention can be found in Kuipers et. al. _Uniform
Distribution of Sequences_.

-Chris

[Ilan Vardi]

The answer is yes, p_i/p_j is dense in R+. I think that this is
a special case of the following lemma with f(x) = x log x:

If a sequence {a_n} is asymptotic to f(n), where f is a continuous
function satisfying

f'(x)/f(x) = o(1)

limsup f(x) = infinity

f'(x) > 0.

then a_i/a_j is dense in R+.

Stanford Sabbatical Problem Solving Group = {Ted Alper, Ilan Vardi}