Which conjectures only need the Grand Riemann Hypothesis to become genuine theorems? 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?
 A: 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.
A: For the Grand Riemann Hypothesis (RH for zeros of all automorphic $L$-functions), see the (somewhat technical) answer to
Equivalent forms of the Grand Riemann Hypothesis
I think the Generalized Riemann Hypothesis (RH for zeros of Dirichlet $L$ functions) has the most significant number theoretic consequences.  In addition to those listed at 
http://en.wikipedia.org/wiki/Generalized_Riemann_hypothesis#Consequences_of_GRH
such as easy primality testing and good bounds on primes in arithmetic progressions, one also gets good lower bounds on class numbers for positive definite binary quadratic forms of discriminant $D$ (or equivalently, rings of integers in complex quadratic fields):  for every $\epsilon>0$ there exists an effective constant $C(\epsilon)$ such that the class number $h(d)>C(\epsilon)|D|^{1/2-\epsilon}$.
A: The main result 
''Assume that the generalized Riemann hypothesis (GRH) for
zeta functions of number ﬁelds holds. There exists a deterministic algorithm that on input positive integers $n$ and $k$, together
with the factorization of $n$ into prime factors, computes the element $T_n$ of the Hecke algebra $T(1, k)$ in running time polynomial in $k$ and $\log n$.''
of the recent book by Couveignes, Edixhoven, et al. (page 3)
http://www.math.univ-toulouse.fr/~couveig/book.htm
assumes the generalized Riemann hypothesis.
A: Consult the chapter entitled
Assuming the Riemann Hypothesis and Its Extensions … on pages  61--67 of the recent book
The Riemann Hypothesis:
A Resource for the Afficionado and Virtuoso Alike by 
Peter Borwein, Stephen Choi, Brendan Rooney and Andrea Weirathmueller.
