(To expand on this a bit: the reason that GRH is so important is because of its ubiquitous applications in mathematics. Maybe you can think of these application

(To expand on this a bit: the reason that GRH is so important is because of its ubiquitous applications in mathematics. Maybe you can think of these applications as different food items, whose recipes call for GRH. The problem is that GRH is a very hard ingredient to come by! (That is, if you want your result to be "unconditional," in the sense of not assuming anything unproven.) In some of these cases, however, (such as the two mentioned in the comment above), substituting the Generalized Lindelof Hypothesis in place of GRH (remarkably) would give you a result just as good! (Think "I can't believe it's not GRH!") Unfortunately, Lindelof is also presently out of reach in almost all contexts. Subconvexity estimates are partial progress towards Lindelof. If you substitute them in place of GRH in your recipes, you'll get something better than if you used the cheapest ingredient (the convexity bound), but not something quite as good as if you had used Lindelof or GRH. Some of the most important results in analytic number theory are "I can't believe it's not GRH!" in their respective contexts. Here is one such example: For applications in sieve theory (e.g., the 2013 breakthrough work on bounded gaps in primes), the Bombieri-Vinogradov theorem is a substitute for GRH that is just as good as GRH.)