3 ms·

While I had a couple of gripes with the article, I don't get the sense that the article exaggerates the significance of this work. The (Generalized) Riemann Hyp

While I had a couple of gripes with the article, I don't get the sense that the article exaggerates the significance of this work. The (Generalized) Riemann Hypothesis implies the (Generalized) Lindelof Hypothesis. The latter is a particularly important consequence of GRH. Not only is it worthwhile, if you are interested in GRH, to verify its predictions (if you proved one of the predictions false, then you would disprove GRH, and if you proved it true, this would in some sense provide "theoretical support" for GRH), but Lindelof is useful in is own right. In particular, it can sometimes serve as a substitute for GRH, and results that can substitute for GRH (such as Bombieri-Vinogradov, zero-density estimates, and subconvexity estimates) are very valuable. Contexts where subconvexity estimates can serve in place of GRH include Quantum Unique Ergodicity, and representations of integers by ternary quadratic forms. For this reason, this paper is a much bigger deal (in the analytic number theory world / in my opinion) than the paper a couple years back that made big headlines about Jensen polynomials (and Nelson's paper is surely one of the biggest results in the last couple of years in analytic number theory). While the latter was more of a "curiosity" / cute paper on an obscure way of thinking about RH, Nelson's paper is big progress on a well-studied weaker form of GRH with significant utility for applications. (I mean, of course, applications in mathematics.)

gavagai691 5y ago
(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.)

gavagai691 5y ago

(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.)