Hello,
I've been interested in number theory for several years, and as time goes by, I read more and more articles in which theorems begin with "Assume the Riemann Hypothesis holds." But up to now, I think I've almost never seen any beginning with "Assume the Grand Riemann Hypothesis holds". So, which are those "theorems" that only need the Grand Riemann Hypothesis to become certain results?
Best Answer
I like the phrase "only need the grand Riemann hypothesis"...
One of my favorite results known contingent on this result (rather, the weaker generalized Riemann hypothesis), is that the ring of integers in a number field (EDIT: with infinite unit group) is Euclidean with respect to some Euclidean algorithm if an only if is is a PID. Interestingly, the "amount" of GRH needed here far exceeds that of the field in question. One must assume GRH for an infinite number of extension fields as well.