3 ms·

I love how every quote by the mathematician clearly states that this does not bring us appreciably closer to the Riemann hypothesis, or that it is very unlikely

by zwkrt 5y ago

I love how every quote by the mathematician clearly states that this does not bring us appreciably closer to the Riemann hypothesis, or that it is very unlikely, while the bulk of the article seems to imply otherwise.

gavagai691 5y ago
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.)
charlieyu1 5y ago
Scientific journalism. Buzzwords, speculations that don’t sound plausible to the scientific community, etc
- Ericson2314 5y ago
  I don't think that's fair. Much scientific journalism, by focusing on exciting unexpected results, leaves one with an impression that there is too much dogma and perhaps other errors. Math is very hard, but has the luxury of correctness being better defined and easier to verify than about any other field. There simply isn't as much room for conspiracy thinking. And accordingly, the articles in Quanta Magazine are both great to read and don't have a side effect of causing any outsider smugness. That isn't to say math doesn't have controversies. The utility of proof assistance and degree to which people should study and use unifying frameworks like Category Theory are examples of ongoing controversies. But it's very fortunate that to be immune to issues like widespread replication crises. Math is really really different than Science.
  - SantalBlush 5y ago
    I'm not aware of category theory being controversial. A much better example would be Mochizuki's IUT controversy.
    - Ericson2314 5y ago
      It's not controversial in terms of correctness, but it is still a bit controversial in terms of whether it's worth the effort? See all those "but does this help me get results for existing problems?" type questions. There have always been mathematicians that prefer to first establish more structure vs those that just want to get down to business. Category theory especially in the past really pushed the envelope on the "more structure" side of things, inflaming those old tensions.
pelasaco 5y ago
probably different than the author from the article, the Mathematician wasn't interested in generating clicks.
deleted 5y ago
[deleted]

by zwkrt 5y ago

gavagai691 5y ago

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.)

charlieyu1 5y ago

Scientific journalism. Buzzwords, speculations that don’t sound plausible to the scientific community, etc

Ericson2314 5y ago
I don't think that's fair. Much scientific journalism, by focusing on exciting unexpected results, leaves one with an impression that there is too much dogma and perhaps other errors. Math is very hard, but has the luxury of correctness being better defined and easier to verify than about any other field. There simply isn't as much room for conspiracy thinking. And accordingly, the articles in Quanta Magazine are both great to read and don't have a side effect of causing any outsider smugness. That isn't to say math doesn't have controversies. The utility of proof assistance and degree to which people should study and use unifying frameworks like Category Theory are examples of ongoing controversies. But it's very fortunate that to be immune to issues like widespread replication crises. Math is really really different than Science.
- SantalBlush 5y ago
  I'm not aware of category theory being controversial. A much better example would be Mochizuki's IUT controversy.
  - Ericson2314 5y ago
    It's not controversial in terms of correctness, but it is still a bit controversial in terms of whether it's worth the effort? See all those "but does this help me get results for existing problems?" type questions. There have always been mathematicians that prefer to first establish more structure vs those that just want to get down to business. Category theory especially in the past really pushed the envelope on the "more structure" side of things, inflaming those old tensions.

pelasaco 5y ago

probably different than the author from the article, the Mathematician wasn't interested in generating clicks.

deleted 5y ago

[deleted]