TITLE: Which conjectures only need the Grand Riemann Hypothesis to become genuine theorems?
QUESTION [5 upvotes]: 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?

REPLY [3 votes]: 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.