On Tue, 10 Jun 2003, Richard Schroeppel posted messages from Victor Miller:
The well-known structure theorem for finite abelian groups (due, I found, to Kronecker) says that to every such group there are a unique set of elementary divisors 1 ne e_1 | e_2 | ... | e_t. The question is what is t -- the number of elementary divisors -- called? I had been calling it the "rank", but that can't be right, since rank is reserved for the number of generators of infinite order in a finitely generated abelian group. Alternatively t is also max_p dim (A/pA), where dim denotes the vector space dimension over GF(p).
It's usually called the torsion rank, and the torsion rank of the Sylow-p-subgroup is the p-rank. The torsion rank is the maximal p-rank over all primes p. There's some ambiguity about what "rank" means in the more general case of a finitely generated abelian group - is it the number of cyclic factors or the number of infinite ones? Privately, I call the former one the "total rank", and the latter the "dimension".
As pointed out by others, even assuming the Riemann Hypothesis, one can't prove that intervals like [x,x+C \sqrt{x}] contain any primes. However, there are positive results in this area.
You omitted to note (maybe because others already had?) the fact that the inf of the theta for which [x, x + x^theta] always contains primes is the inf of the sigma for which the strip sigma < Re(z) < 1 is free of zeros of the zeta-function. So the Riemann hypothesis is equivalent to the assertion that this inf is 1/2. John Conway