Victor Miller accidentally sent these to the old Math-Fun address. I disabled forwarding to the new address a few months ago, because of too much spam. The posting address is math-fun@mailman.xmission.com Rich ---------------------- To: math-fun@CS.Arizona.EDU Subject: Name for a classical invariant From: victor@idaccr.org (Victor S. Miller) Date: Mon, 09 Jun 2003 16:49:05 -0400 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). Victor ---------------------- To: math-fun@CS.Arizona.EDU Subject: Re: [math-fun] a prime in each row? From: victor@idaccr.org (Victor S. Miller) Date: Tue, 10 Jun 2003 12:11:03 -0400 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. One of the most interesting is by Pat Gallagher, "On the distribution of primes in short intervals", Mathematika v. 23 (1976) pp 4-9. In this paper he proves that if one assumes a uniform version of the Hardy-Littlewood prime d-tuple conjecture (given d_1, ..., d_r, there is a constant C_d such that the number of n <= X for which n+d_1, ..., n+d_r are all prime is asymptotic to C_d X/(log X)^r) that prime gaps are Poisson distributed. That is, if P_k(h,N) = the number of intervals [n,n+h] with n <= N containing exactly k primes, then P_k(h,N) is asymptotic to N exp(-\lambda) \lambda^k/k! where h is asymptotic to \lambda log(N). There are also results (I forget where right now), that if one defines an x to be "bad" if [x,x+sqrt{x}] contains fewer than 0.5 * sqrt{x}/log(x) primes, then the number of bad x <= X is o(X) (there's something more specific known here, but I don't recall what it is). -- Victor S. Miller | " ... Meanwhile, those of us who can compute can hardly victor@idaccr.org | be expected to keep writing papers saying 'I can do the CCR, Princeton, NJ | following useless calculation in 2 seconds', and indeed 08540 USA | what editor would publish them?" -- Oliver Atkin ---------------------- To: math-fun@CS.Arizona.EDU Subject: Re: [math-fun] a prime in each row? From: victor@idaccr.org (Victor S. Miller) Date: Tue, 10 Jun 2003 12:28:26 -0400 I should also add that a great paper to read on this is by Andrew Granville "Harald Cramer and the distribution of prime numbers" available at http://www.dartmouth.edu/~chance/chance_news/for_chance_news/Riemann/cramer.... It is also published in the Scandanavian Actual Journal, but that seems a bit obscure. It gives a comprehensive survey of what's known in the field about distribution of primes in short intervals. -- Victor S. Miller | " ... Meanwhile, those of us who can compute can hardly victor@idaccr.org | be expected to keep writing papers saying 'I can do the CCR, Princeton, NJ | following useless calculation in 2 seconds', and indeed 08540 USA | what editor would publish them?" -- Oliver Atkin