Two inaccuracies in the article. For purposes of simplicity, the author writes "But the Lindelöf hypothesis says that as the inputs get larger, the size of the output is actually always bounded at 1% as many digits as the input." But this is a case where simplification goes too far. What Lindelof says is that the size of the output is always bounded by ε% as many digits as the input, for ANY (arbitrarily miniscule) ε > 0.
Second, the subtitle "Paul Nelson has solved the subconvexity problem..." is strange. The subconvexity problem, for a given L-function, is to lower the percentage described above from 25% to "any positive number"; in other words, to bridge the gap between the convexity bound (which is "trivial") and Lindelof (a consequence of GRH). The only way that the statement "Paul Nelson has solved THE subconvexity problem..." could maybe be accurate is if Nelson proved the Lindelof hypothesis for "all" L-functions. Which is far from the case. (What makes subconvexity so interesting, as Nelson says in the article, is that it is a problem where you can make partial progress towards a particularly important consequence of GRH. And Nelson's result is exactly that: partial progress.) More accurate would be "Paul Nelson has made significant progress on the general subconvexity problem."
These slips from Quanta are puzzling. Quanta normally seems to be more meticulous about accuracy, and the author is a veteran.
The attraction of Quanta has always been that its writers seemed to have an intrinsic passion for math, treating it as more than just a gee-whiz topic to gloss over in between the latest panicked missives about politics and how cellphones are destroying our children. One of its most delightful qualities has been the consistent willingness to take a few extra sentences to explain even advanced concepts in a manner that is basically technically correct.
Certainly the article remains far better than the average mass media STEM writing, but Quanta should take greater care to keep their quality pristine. They have been utterly unique and have set the example for everyone else.
It's not entirely fair to call the second point a major slip. I think that it is still sort of inaccurate, but not exactly for the reason I had said above; see the thread following kevinventullo's comment below. In any case, my second point is sort of pedantic. I don't love the terminology used, but I wouldn't heap very much blame on the author.
The first one is a bit worse. I think if I had read this knowing nothing about convexity I would have gotten the wrong idea from the arbitrary choice of 1%. I understand the desire to simplify, but it is an art to simplify while keeping what you say technically correct. Quanta usually does an excellent job of this. I wouldn't say that the first point above is an egregious error by any means, but I think it is a slip.
Ian Petrow defines the Subconvexity Problem as "Prove non-trivial upper bounds for L-functions on the critical line" [1]
Given that, it seems fair-ish to say that Nelson solved the Subconvexity Problem. You just have to understand that the problem is really a family of problems of increasing hardness (e.g. prove tighter and tighter bounds), and solutions more powerful than Nelson's may come later.
By that definition, I agree, if you add the qualifier "Nelson solved the subconvexity problem for for automorphic L-functions."
I still think it's vague to talk about "the" subconvexity problem without specifying what variable you want the bound to be subconvex in, but, really, who am I to argue with Ian Petrow..!
Over the past few years, Quanta has become my go-to place for learning about current science developments. A decade or more Scientific American started focusing on their political agenda. And in the last year or two, American Scientist has been completely taken over by wokeness. I haven't found a better venue for pleasure-type learning about science than Quanta.
To be fair, I think there are economic conditions that drive the unfortunate situation. Quanta is very lucky to have enormous free support from a billionaire-funded foundation (Simons Foundation).
When publications become desperate for cash, they dive into the politics and culture war rathole.
I’m confused by your second paragraph. The article describes the subconvexity problem for a given L-function as lowering the bound from 25% to x% for some x < 25. That is, 25% is the “convexity bound” and the existence of such an x is a “subconvexity bound.”
My understanding is that Nelson has indeed done this for every L-function, although the exact x varies with the L-function. How is that not solving the subconvexity problem?
A subconvex bound is any bound that beats the convexity bound of 25%. If you would define "the subconvexity problem" as beating the 25% bound AT ALL (or in other words providing any subconvex bound), then the statement in the subtitle would be closer to accurate.
But this is not (I think) how most people would speak, and I wouldn't say this. I would say the subconvexity problem is the problem of bridging the gap between 25% (convexity) and 0% (Lindelof). Subconvexity bounds for various L-functions (zeta, Dirichlet L-functions, and higher rank L-functions) are a very active area of research, and so it seems strange to me to say "the subconvexity problem is solved."
"My understanding is that Nelson has indeed done this for every L-function." One minor note. Nelson's work applies if the L-function is automorphic. By the Langlands philosophy, this should be true for any reasonable L-function, but this is far from known, even in, e.g., the simple case of the L-function corresponding to the tensor product of two automorphic L-functions.
Edit: I am wrong about the statement "this is not (I think) how most people would speak." It looks like beating the convexity bound at all is often described as "solving the subconvexity problem," e.g. in the introduction here https://link.springer.com/article/10.1007/s002220200223. This description is a bit strange to me, but if it is standard then it is unfair for me to call it an "inaccuracy," persay; thanks for pointing this out.
As a further addendum, there are different "aspects" in which you could want the bound to be subconvex. As a result, even for a given family of L-functions) there is more than one "subconvexity problem." In this case, Nelson's bound is not subconvex in the non-Archimedean aspect. This is another reason why it doesn't seem quite right to me to say without qualification that "the subconvexity problem has been solved."
As you say, we now know that each standard L-function satisfies a subconvex estimate as its argument varies. This falls short of "solving the subconvexity problem" in two respects.
The first, pointed out already by gavagai691, is that it is not known that "every L-function" is "standard" (this is a major open problem in the Langlands program).
The second is that the general "subconvexity problem" asks for a nontrivial estimate not only for each fixed L-function as its argument varies, but more generally as the L-function itself varies. The general case of the problem remains open.
> What Lindelof says is that the size of the output is always bounded by ε% as many digits as the input, for ANY (arbitrarily miniscule) ε > 0.
Is there are reason or benefit of stating it like that rather than saying the fraction of output and input digit size asymptotically approaches zero? Or did I misunderstand the explanation?
Thinking about digit size is sort of fine for simplification / not turning off general audiences with equations, but a bit of a pain to think about if , e.g., the things you're comparing might be smaller than one.
What Lindelof says more precisely is that for any ε > 0 and any L-function L(s), there is a constant C_(ε,L) (depending only on ε and L, but crucially not on s) such that |L(s)| <= C_ε (1+|Im(s)|)^ε.
There may be some linguistic benefits to the "epsilon" formulation when discussing subconvexity rather than Lindelof. For instance, I think "the output is eventually bounded by 24% of the input" sounds more natural than "the limsup of output/input approaches something less than 24%".
Second, the subtitle "Paul Nelson has solved the subconvexity problem..." is strange. The subconvexity problem, for a given L-function, is to lower the percentage described above from 25% to "any positive number"; in other words, to bridge the gap between the convexity bound (which is "trivial") and Lindelof (a consequence of GRH). The only way that the statement "Paul Nelson has solved THE subconvexity problem..." could maybe be accurate is if Nelson proved the Lindelof hypothesis for "all" L-functions. Which is far from the case. (What makes subconvexity so interesting, as Nelson says in the article, is that it is a problem where you can make partial progress towards a particularly important consequence of GRH. And Nelson's result is exactly that: partial progress.) More accurate would be "Paul Nelson has made significant progress on the general subconvexity problem."