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I was reading this when I came across Gourevitch's conjecture.

My understanding is that solutions to these series are of practical interest. If one encounters such a series, being able to solve it exactly is more practical than having to solve it numerically.

But, not being a mathematician, I simply can't imagine what the theoretical implications of proving such conjectures are.

What are they?

Andrej Bauer
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CHM
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    The theoretical implication is that this is really cool, and understanding why it is true is likely to require understanding something deep. – Igor Rivin Aug 24 '12 at 20:05
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    Isn't this like porn? – Mariano Suárez-Álvarez Aug 24 '12 at 20:44
  • Mariano: Yes, in so many ways! – Steven Landsburg Aug 24 '12 at 20:54
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    @CHM: It is usually difficult to predict in advance what one will learn. If you want to see a baby example, check out the many proofs that $\sum_{i=1}^\infty \frac{1}{n^2} = \frac{\pi^2}6.$ Every one of them has many interesting ideas. – Igor Rivin Aug 24 '12 at 21:27
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    Voted to close. As you seem to be an undergraduate in chemistry or related, it is unlikely we can convince you of the value of this, and I don't think anyone should try. If you find further questions, try http://math.stackexchange.com/questions – Will Jagy Aug 24 '12 at 21:49
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    Not sure that exactly is necessarily more practical than numerical - try $\zeta(100)$? –  Aug 24 '12 at 21:51
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    @Carl: $\zeta(100) = 1, $ to machine precision. – Igor Rivin Aug 24 '12 at 21:54
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    @WillJagy How absurd, pejorative and close minded. Please tell me you're not serious. – CHM Aug 25 '12 at 01:09
  • @IgorRivin thanks for the suggestions. – CHM Aug 25 '12 at 01:09
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    CHM - Please don't be offended by the votes to close. This is not personal! Mathoverflow is specifically for questions that arise in mathematical research, as the FAQ makes clear. Your question is a great one, but as you admit is not a question that arose out of mathematical research. You might find that math.stackexchange.com would be a better venue. – HJRW Aug 25 '12 at 07:47
  • @CHM: I find that the question you ask is interesting. But the only way of giving a satisfactory answer is in comparison with other mathematics. For every mathematical result, one can ask "what is the interest?". The answers will vary... they will vary per question, and per mathematician who tries to answer. What mathematicians try to do every day is to "wrap their mind" around difficult concepts: anything you can say about something is progress. Back to your question: I'd say that finding a closed form is interesting, but only to the extent that the infinite series you study is interesting. – André Henriques Aug 25 '12 at 11:19
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    @HW It's the justification that irritated me, not the vote to close. I would accept my question being closed for reasons such as the one you mention. – CHM Aug 25 '12 at 18:26
  • @Mariano, @Steven: I don't understand in what sense it is like porn? – timur Aug 25 '12 at 23:44
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    @timur: I am fairly certain that Mariano makes reference to a fairly well-known quote of some jusdge related to the difficulty of precisely definining it, which in a condensed form says something like: hard to define but I know it when I see it, see http://en.wikipedia.org/wiki/I_know_it_when_I_see_it for details. –  Aug 26 '12 at 01:17
  • To paraphrase something that I said in another MO answer, Gourevitch's conjecture (and conjectures like it) are really implicit challenges to find the hidden structure, and finding hidden structure is a large part of what mathematics is all about. If Gourevitch's conjecture is true (which it surely is) then mathematicians have faith that it is true "for a reason" and not true by accident. The hope is that figuring out the reason will give us new mathematical insights that we previously did not suspect. – Timothy Chow Jul 29 '19 at 00:02
  • (continued) So, depending on what you mean by "practical interest," the identity in Gourevitch's conjecture may not be of "practical interest." I suspect that by "practical interest" you mean that the identity itself is directly used in the proof of (or calculation of) something that is interesting for some other reason. This is possible, but it seems unlikely to me. OTOH, if proving the conjecture yields a new mathematical insight, then that new insight could have all kinds of practical ramifications (but it's hard to say what those might be until we actually obtain the new insight). – Timothy Chow Jul 29 '19 at 00:14

5 Answers5

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Let $k \ge 2$ be an integer. Consider the series $$ \sum_{n=1}^{\infty} \frac{1}{n^k} $$ this is typically denoted $\zeta(k)$ as it is the value of the Riemann zeta function at $k$.

Now, Euler showed that for even $k$ this is equal to $$q_k \pi^k $$ where $q_k$ is some rational number (that one can also describe explictly).

This is on the one hand an interesting fact and would also allow to calculate approximations of the powers of $\pi$ but what I actually want to get at is that from this it follows that if one knows that $\pi$ is transcendental then one gets directly that $\zeta(k)$ is transcendental and in particular not a rational number.

So this is for even $k$. What about odd $k$? Say $k=3$. Is this rational or irrational? This question was open for a long time until it was proved at the end of the 1970s by Apéry.

How does this prove go (very roughly!):

He first showed that $$ \zeta(3) = \frac{5}{2} \sum_{i=1}^{\infty} \frac{ (-1)^n}{n^3 \binom{2n}{n}}.$$

So one could say he evaluated the series on the right in 'a closed form'; showing that its values is something already known/defined.

Then based on this he derived some sequences of rational numbers that converge to $\zeta(3)$ so fast that it is impossible for $\zeta(3)$ to be rational itself thus proving the irrationality of $\zeta(3)$.

So, in order to show that $\zeta(3)$ is irrational he first needed to show that it is equal to the limit of this (other) series, or put differently to evaluate this series; not only the first simpler one.

It would be interesting to be able to do something like this for other odd numbers, but so far no-one knows how to do so and the irrationality of $\zeta$ at any other odd positive integers is unknown (although there are results that assert that among certain collections of them there are at least some irrational ones).

Thus finding an evaluation of a series can be used to infer something theoretical on its value.

This is not always the motivation, but sometimes it is the case that the point is not so much to know the value of the series in order to replace it in some computation say, but rather to use the series as a form of describing its value by simpler building blocks and thereby allowing to learn something (new) on the value.

J. M. isn't a mathematician
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I can recommend reading "Closed forms: what they are and why we care" by Jon Borwein and Richard Crandall, the article is to appear in Notices Amer. Math. Soc. 60 (2013).

Edit (Dec 2012). The paper has just appeared in the Notices: pdf. Together with the sad news about Richard Crandall: he passed away.

Wadim Zudilin
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    Wow. Anyone who's ever asked for a "closed form solution" should read this paper. Very eye-opening on a topic that I think most of us take for granted. – Aeryk Aug 26 '12 at 02:08
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HERE is an example in MO. For the first question, I could evaluate the series in closed form, so I could compute with their known properties, and thus answer the question. For the second question, the series were just ${}_2F_1$ series, and I did not know their properties, so I could not answer the question. Despite the numerical evidence in the graphs.

Community
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Gerald Edgar
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Having the close form for a series of functions is also important:

  1. Closed forms can sometimes be used to find exact solutions to differential equations.
  2. Closed forms allow for accurate estimates on errors in approximations of special functions. Some of these are used in your calculator.
  3. Closed forms can sometimes produce proofs of non-series results. For example, the equation $$ e^{ix} = \cos(x) + i\sin(x) $$ is proven by using three Taylor series.
  4. This is a bit incestuous, but having the closed form for one series allows for manipulation to produce the closed form of other series.
Marc Chamberland
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This is an old question, but just popped up again, so I thought I'd mention a very important situation where explicit formulas are of fundamental theoretical importance. Number theorists are very interested in understanding the ideal class group $H_p$ of the cyclotomic field $\mathbb Q(\zeta_p)$, where $\zeta_p$ is a primitive (prime) root of unity. People use explicit formulas for special values of $L$-series to construct integers that annihilate the class group, thereby obtaining information about the size of the class group. And for elliptic curves, Rubin in the CM case and Kolyvagin for curves over $\mathbb Q$ have used special values such as $L(E,1)$ and $L'(E,1)$ to prove that the Tate-Shafarevich group of $E$ is finite, which had been a long-standing conjecture. Very (very) roughly, they use the special values to construct an integer $m$ that annihilates every element of SHA, and from there the finiteness follows, since it is not hard to prove that SHA$[m]$ is finite. This is the theory of Euler Systems, which is still attracting lots of research attention.

I realize that $L$-series are Dirichlet series, and this question asks about "infinite series", which is probably meant to refer to power series. But one can move back and forth between $L$-series and power series using the Mellin transform, and indeed, for elliptic curves over $\mathbb Q$, the first step in studying $L(E,1)$ is to write it as the Mellin transform of a modular form, courtesy of Wiles et al.

Joe Silverman
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