I have a question for the mathematical community about the language mathematicians use for a series of functions, and it has been bugging me for a while. I was thinking of posting it on MathOverflow, but it seems to fit the “subjective and argumentative” criterion for closing rather well.

Suppose someone introduces a countably infinite set of holomorphic functions $\{ f_n \}_{n \geq 0}$ on some open subset U of the complex numbers, and wants to argue that the sequence sum converges to a holomorphic function. One valid way to prove it would be to show that the sequence sum converges locally uniformly absolutely (or uniformly absolutely on compact subsets of U), and then point toward the complex analysis text of choice.

Question: If someone only argues that the sequence sum converges absolutely, should I complain? (If so, how loudly?)

I tend to think that absolute convergence is a pointwise statement, and doesn’t a priori tell me anything about holomorphicity. On the other hand, there is a theorem of Osgood (Ann. Math. 1901) stating that pointwise convergence of a sequence of holomorphic functions to a function implies that there is an open dense subset V of U, such that the function is holomorphic on V, and the convergence is uniformly absolute on compact subsets of V. While this theorem is rather impressive, it doesn’t give me quite what I want, since V is not necessarily all of U (and there are interesting counterexamples, e.g., in this expository pdf). Unfortunately, a sample of published papers that use holomorphic functions reveals that people use the absolute convergence language a lot. This leads me to consider the following two (non-exclusive) possibilities:

1. The term “absolute convergence” has become an abbreviated form of “uniformly absolute convergence on compact sets” among people who know what they are doing.
2. Some mathematicians are rather sloppy when discussing convergence, and this includes referees.

Any input would be greatly appreciated, especially input from those more analytically experienced than I.

[Apologies for any confusion caused by the previous version.]

## 14 thoughts on “Question about convergence norms”

1. Back when I did analysis I routinely used dozens of topologies on function spaces, and I never heard anyone talk about a sequence of functions converging absolutely.

Uniformly, sure! Locally uniformly, sure! There are all sorts of nice theorems in complex analysis about uniformly or locally uniformly convergent sequences of functions. But I don’t even know what it means for a sequence of functions or numbers to converge absolutely.

I know what it means for a series or integral to converge absolutely: it means that it still converges when you stick an absolute value sign around the summand or integrand. But why would I want to stick an absolute value sign inside a limit?

The expository paper you linked to, by Steven Krantz, doesn’t seem to contain the words “absolute” or “absolutely”.

So, if you’re ever tempted to talk about an absolutely convergent sequence of functions or numbers, please at least explain what you mean by it, because at least one ex-analyst out there has no idea what it means.

2. Sorry, please replace my last mention of the word “series” by “sequence”.

[Done! DES]

3. Oops, I oversimplified my situation! I’m actually looking at a sum of functions, so pointwise absolute convergence is a meaningful statement. I shall change the post appropriately.

4. GS says:

Sure, absolute convergence is a pointwise property. However when people say “so-and-so series converges absolutely in this region”, they also give some estimate to show that this is true. And this estimate would also show that it is actually uniformly convergent on every compact subset.

I think no harm is done and it is safe to omit an explicit statement.

5. GS, I am having difficulty parsing your statement. Are you claiming that everyone gives such estimates when they make claims of convergence, or are you saying that giving such estimates without using the word “uniform” is an ideal situation? I certainly have many counterexamples to the first interpretation.

Perhaps there is a third interpretation: it is okay to say “absolute convergence” if such an assertion is accompanied by an estimate that proves local uniformity, but otherwise it is not okay. Is that what you mean?

6. Okay, thanks, now I think I understand.

As an ex-analyst, I say: never be sloppy when describing the sense in which some series converges.

If you’ve got a sum of functions that converges locally uniformly, say so – please don’t make your poor reader stare at your estimates and deduce that this fact follows from those estimates. Ideally your reader should be able to skip the estimates, if they’re willing to trust you, and still understand what you’re saying, just by reading your prose. English is a wonderful thing.

And please don’t say a sum of functions converges “absolutely” if what you really mean – or need – is that it converges locally uniformly. I don’t see how it helps to be inaccurate in this way.

7. Thank you, John. I was leaning in this direction, but bringing up the reader’s perspective was very helpful.

Now I am wondering if this is just a “term of art” among analytic number theorists…

8. DC says:

(2) is very common in my personal experience, especially if some interesting specifics are at issue and the general analysis is not “the point.” what i mean by this vague phrase is for example if the holomorphic functions are “coming from” something that gives them extra “structure”, and nobody looking at problem X is studying holomorphic functions that don’t arise in that way, people, including referees, get sloppy. if one is not considering the totality of all conceivable holomorphic functions when making statements about them, one gets lazy.

i do not think it is a matter of specialists skipping routine arguments for brevity. as examples like the paper you point to indicate, there is very little in the way of an abstract body of general convergence tools that people might point to but don’t. they just avoid the issue by not touching too heavily upon it. sometimes this is excusable (maybe even most of the time). it is a judgment call.

9. Without having concrete examples of sloppy papers to look at, it’s a bit hard to comment precisely, but in analytic number theory, people use constantly without reference the lemma that says that if a Dirichlet series converges at a single point, then it converges absolutely locally uniformly (hence is holomorphic) at all points with (strictly) larger real part. This is typically proved only in the introductory textbooks (e.g., it is in Serre’s “Course in arithmetic”), and research papers will not mention it. (This is similar to the existence of radius of convergence of a power series).

10. GS says:

#5. What I had in mind is the following. Whenever I saw a statement like “this series is absolutely convergent in this region and is therefore holomorphic”, it was accompanied by some estimate to show the absolute convergence. And it was always the case that this same estimate showed uniform convergence in every compact subset. For example, the convergence of $L$-functions prof. Kowalski is talking about, is such an experience.

I should have added that in my limited experience, I had no difficulties in supplying the extra “uniformly convergent in every compact subset” part very easily using the same estimate for showing absolute convergence. I am not qualified to say anything more than this. Of course, if there are statements when an estimate is not given, then my statement in #4 does not make sense. I do not know what to do in such cases.

11. Thank you Emmanuel; that was a good example. I was afraid there would be some subtlety regarding what theorems the cultured reader is expected to know. I guess it is unreasonable for me to be completely dogmatic on this point.

12. Pete L. Clark says:

#9: I am a little confused about your remark that Dirichlet series converge “absolutely locally uniformly” in the half-plane to the right of any point of convergence. The Dirichlet series associated to a non-trivial Dirichlet character converges for Re(s) > 0 but converges absolutely only for Re(s) > 1. I know you know this (I learn analytic number theory by reading your books), so either I am misunderstanding you or it is just a typo.

Concerning Scott’s question: I think there is definitely a history in, say, analytic number theory, of experts writing for other experts and not having to explain themselves to a more general mathematical audience. (My pet peeve in watching analytic number theory talks is that the quantifiers are often omitted — or even present but incorrect! I think the experts have a deep understanding of which things are fixed with respect to which other things which can transcend their ability to communicate this clearly to others.) I am pretty sure that most leading analysts know far better than to assert holomorphicity in a situation where it in fact does not exist: in other words, they are making mistakes of exposition, at worst. However, the problem is that the assumed unwritten knowledge of one generation of experts can be lost by later readers who can see only what is on the page.

13. #12: speak of sloppy writing… I actually misremembered the actual statement, which is that if a Dirichlet series converges at s_0, then it converges uniformly (but not necessarily absolutely) in any angle with vertex Re(s_0) in the right-hand plane Re(s)>Re(s_0). In particular, it converges locally uniformly in that half-plane, hence it defines a holomorphic function there. (This is proved by Abel summation to check the Cauchy criterion, and I think that’s the statement in Serre’s book).

I was confusing this with the easier fact that if a Dirichlet series converges at s_0, then it converges _absolutely_ at any point with Re(s)>s_0+1. (The +1 that shows up in the case of Dirichlet series), which is because

$|a_n|n^(-Re(s_0)-1-\delta)\leq Cn^{-1-\delta)$

since the convergence at s_0 implies that the a_nn^(-s_0) are bounded in modulus.

Maybe the best way to phrase this is the statements that a Dirichlet series converges in a open half-plane (possibly empty), and defines a holomorphic function in this region, and converges absolutely in a (possibly different) half-plane, but is at most one unit to the right of the first one.

As for the general problems of exposition of analytic number theory compared with other fields, I think this is a general effect of specialization. I’ve been lost and confused in many talks in algebraic or analytic number theory, in algebraic geometry and arithmetic geometry, not to mention algebra or pure analysis. I’ve heard complaints from analytic number theorists that algebraists are unable to motivate their lofty constructions and use big words to frighten the audience, and from algebraists that analytic number theorists do not make clear which quantities are important, and which depend on which others, and why we should care…

It is a hard problem, and often depressing.

14. Pete L. Clark says:

For those who have read #13 and want an immediately available discussion of convergence of Dirichlet series, see e.g. http://math.uga.edu/~pete/4400dirichlet.pdf

If you’re serious, do see Serre’s _A Course in Arithmetic_. For instance I do not prove the uniform convergence in a “Stol[t?]z angle”, the fundamental result from which the holomorphicity follows in this case.

Yes, it’s true that arithmetic geometers can give bad talks too, and that our talks are bad in different ways! For sheer impenetrability arithmetic geometry can be hard to beat. At my institution we have a postdoc who works at the interface of analysis, analytic number theory, and combinatorics. She attends the number theory seminar (which is in fact called the “number theory / arithmetic geometry seminar” and the latter title is most often the more descriptive one) often, but has had some experiences of complete impenetrability, so now I try to give her a heads up as to whether I think a given talk will be one that she can get something out of or whether it’s just a waste of her time. In contrast, I have never seen a talk by a leading analytic number theorist that I felt was a complete waste of my time — at least I can get a sense of what problems they are working on and what techniques they are using to solve them.