The 5040 in the title refers to the largest known counterexample to a bound for the sum-of-divisors function
\[
\sigma(n) = \sum_{d | n} d = n \sum_{d | n} \frac{1}{d} \]
In 1983, the french mathematician Guy Robin proved that the Riemann hypothesis is equivalent to
\[
\frac{\sigma(n)}{n~log(log(n))} < e^{\gamma} = 1.78107... \]
when $n > 5040$.
The other known counterexamples to this bound are the numbers 3,4,5,6,8,9,10,12,16,18,20,24,30,36,48,60,72,84,120,180,240,360,720,840,2520.
In Baez’ post there is a nice graph of this function made by Nicolas Tessore, with 5040 indicated with a grey line towards the right and the other counterexamples jumping over the bound 1.78107…
Robin’s theorem has a remarkable history, starting in 1915 with good old Ramanujan writing a part of this thesis on “highly composite numbers” (numbers divisible by high powers of primes).
His PhD. adviser Hardy liked his result but called them “in the backwaters of mathematics” and most of it was not published at the time of Ramanujan’s degree ceremony in 1916, due to paper shortage in WW1.
When Ramanujan’s paper “Highly Composite Numbers” was first published in 1988 in ‘The lost notebook and other unpublished papers’ it became clear that Ramanujan had already part of Robin’s theorem.
Ramanujan states that if the Riemann hypothesis is true, then for $n_0$ large enough we must have for all $n > n_0$ that
\[
\frac{\sigma(n)}{n~log(log(n))} < e^{\gamma} = 1.78107... \]
When Jean-Louis Nicolas, Robin's PhD. adviser, read Ramanujan's lost notes he noticed that there was a sign error in Ramanujan's formula which prevented him from seeing Robin's theorem.
Nicolas: “Soon after discovering the hidden part, I read it and saw the difference between Ramanujan’s result and Robin’s one. Of course, I would have bet that the error was in Robin’s paper, but after recalculating it several times and asking Robin to check, it turned out that there was an error of sign in what Ramanujan had written.”
For starters, it has been verified for all numbers smaller that $10^{10^{13}}$…
It has been verified, unconditionally, for certain classes of numbers:
all odd integers $> 9$
all numbers not divisible by a 25-th power of a prime
Rings a bell? Here’s another hint:
According to Xiaolong Wu in A better method than t-free for Robin’s hypothesis one can replace the condition of ‘not divisible by an N-th power of a prime’ by ‘not divisible by an N-th power of 2’.
Further, he claims to have an (as yet unpublished) argument that Robin’s inequality holds for all numbers not divisible by $2^{42}$.
So, where should we look for counterexamples to the Riemann hypothesis?
What about the orders of huge simple groups?
The order of the Monster group is too small to be a counterexample (yet, it is divisible by $2^{46}$).
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In April my Google+ account will disappear. Here I collect some G+ posts, in chronological order, having a common theme. Today, math-history (jokes and puns included).
September 20th, 2011
Was looking up pictures of mathematicians from the past and couldn’t help thinking ‘Hey, I’ve seen this face before…’
Leopold Kronecker = DSK (2/7/2019 : DSK = Dominique Strauss-Kahn)
Adolf Hurwitz = Groucho Marx
June 2nd, 2012
The ‘Noether boys’
(Noether-Knaben in German) were the group of (then) young algebra students around Emmy Noether in the early 1930’s. Actually two of them were girls (Grete Hermann and Olga Taussky).
The picture is taken from a talk Peter Roquette gave in Heidelberg. Slides of this talk are now available from his website.
In 1931 Jacques Herbrand (one of the ‘Noether boys’) fell to his death while mountain-climbing in the Massif des Écrins (France). He was just 23, but already considered one of the greatest minds of his generation.
He introduced the notion of recursive functions while proving “On the consistency of arithmetic”. In several texts on Herbrand one finds this intriguing quote by Chevalley (one of the first generation Bourbakis):
“Jacques Herbrand would have hated Bourbaki” said French mathematician Claude Chevalley quoted in Michèle Chouchan “Nicolas Bourbaki Faits et légendes” Edition du choix, 1995. («Jacques Herbrand aurait détesté Bourbaki» in the original French version).
+Pieter Belmans (re)discovered a proto-drawing of Mumford’s iconic map of Spec(Z[x]) in his ‘red book’.
The proto-pic is taken from Mumford’s ‘Lectures on curves on an algebraic surface’ p.28 and tries to depict the integral projective line. The set-up is rather classical (focussing on points of different codimension) whereas the red-book picture is more daring and has been an inspiration for generations of arithmetical geometers.
Still there’s the issue of dating these maps.
Mumford himself dates the P^1 drawing 1964 (although the publication date is 66) and the red-book as 1967.
Though I’d love to hear more precise dates, I’m convinced they are about right. In the ‘Curves’-book’s preface Mumford apologises to ‘any reader who, hoping that he would find here in these 60 odd pages an easy and concise introduction to schemes, instead becomes hopelessly lost in a maze of unproven assertions and undeveloped suggestions.’ and he stresses by underlining ‘From lecture 12 on, we have proven everything that we need’.
So, clearly the RedBook was written later, and as he has written in-between his master-piece GIT i’d say Mumford’s own dating is about right.
Still, it is not a completely vacuous dispute as the ‘Curves’ book (supposedly from 1964 or earlier) contains a marvelous appendix by George Bergman on the Witt ring which would predate Cartier’s account…
Thanks to +James Borger i know of George’s take on this
“I was a graduate student taking the course Mumford gave on curves and surfaces; but algebraic geometry was not my main field, and soon into the course I was completely lost. Then Mumford started a self-contained topic that he was going to weave in — ring schemes — and it made clear and beautiful sense to me; and when he constructed the Witt vector ring scheme, I thought about it, saw a nicer way to do it, talked with him about it and with his permission presented it to the class, and eventually wrote it up as a chapter in his course notes.
I think that my main substantive contribution was the tying together of the various prime-specific ring schemes into one big ring scheme that works for all primes. The development in terms of power series may or may not have originated with me; I just don’t remember.”
which sounds very Bergmannian to me.
Anyway I’d love to know more about the dating of the ‘Curves’ book and (even more) the first year Mumford delivered his Red-Book-Lectures (my guess 1965-66). Thanks.
Pieter maintains an “Atlas of this picture” here
“Serre first considered the set of maximal ideals of a commutative ring A subject to certain restrictions. Martineau then remarked to him that his arguments remained valid for any commutative ring, provided one takes all prime ideals instead of only maximal ideals. I then proposed a definition of schemes equivalent to the definition of Grothendieck. In my dissertation I confined myself to a framework similar to that of Chevalley, so as to avoid an excessively long exposition of the preliminaries!”
In the 1956/57 Chevalley seminar Cartier gave the first two talks and in the first one, on november 5th 1956, one finds the first published use of the word ‘scheme’, which he refers to as ‘schemes in the sense of Chevalley-Nagata’. On page 9 of that talk he introduces the prime spectrum with its Zariski topology.
In the second talk a week later, on november 12th, he then gives the general definition of a scheme (as we know it, by gluing together affine schemes and including the stalks).
BUT, he did all of this ‘only’ for affine rings over a field, ‘to avoid an excessively long exposition of the preliminaries’…
Grothendieck then made the quantum-leap to general commutative rings.
June 18th, 2013
Correction : scheme-birthday = december 12th, 1955
Claude Chevalley gave already two talks on ‘Schemes’ in the Cartan-Chevalley seminar of 1955/56, the first one on december 12th 1955, the other a week later.
Chevalley only considers integral schemes, of finite type over a field (Cartier drops the integrality condition on november 5th 1956, a bit later Grothendieck will drop all restrictions).
Grothendieck’s quote “But then, what are schemes?” uttered in a Parisian Cafe must date from that period. Possibly Cartier explained the concept to him. In a letter to Serre, dated december 15th 1955, Grothendieck is quite impressed with Cartier:
“Cartier seems to be an amazing person, especially his speed of understanding, and the incredible amount of things he reads and grasps; I really have the impression that in a few years he will be where you are now. I am exploiting him most profitably.”
June 18th, 2013
David Mumford on the Italian school of Geometry
Short version:
Castelnuovo : the good
Severi : the bad
Enriques : the ugly
The longer version:
“The best known case is the Italian school of algebraic geometry, which produced extremely good and deep results for some 50 years, but then went to pieces.
There are 3 key names here — Castelnuovo, Enriques and Severi.
C was earliest and was totally rigorous, a splendid mathematician.
E came next and, as far as I know, never published anything that was false, though he openly acknowledged that some of his proofs didn’t cover every possible case (there were often special highly singular cases which later turned out to be central to understanding a situation). He used to talk about posing “critical doubts”. He had his own standards and was happy to reexamine a “proof” and make it more nearly complete.
Unfortunately Severi, the last in the line, a fascist with a dictatorial temperament, really killed the whole school because, although he started off with brilliant and correct discoveries, later published books full of garbage (this was in the 30’s and 40’s). The rest of the world was uncertain what had been proven and what not. He gave a keynote speech at the first Int Congress after the war in 1950, but his mistakes were becoming clearer and clearer.
It took the efforts of 2 great men, Zariski and Weil, to clean up the mess in the 40’s and 50’s although dredging this morass for its correct results continues occasionally to this day.” (David Mumford)
June 19th, 2014
Hirune Mendebaldeko – Bourbaki’s muse
After more than 70 years, credit is finally given to a fine, inspiring and courageous Basque algebraic geometer.
One of the better held secrets, known only to the first generation Bourbakistas, was released to the general public in april 2012 at the WAGS Spring 2012, the Western Algebraic Geometry Symposium, held at the University of Washington.
Hirune Mendebaldeko was a Basque pacifist, a contemporary of Nicholas Bourbaki, whom she met in Paris while there studying algebraic geometry. They were rumored to be carrying on a secret affair, with not infrequent trysts in the Pyrenees. Whenever they appeared together in public, however, there was no indication of any personal relationship.
From the comments, by +Sandor Kovacs:
+Chris Brav Chris, just between us: the whole thing is a joke. I just tried to put yet another twist on it. Also, until now we have never admitted that it is, so please don’t tell anyone. 😉
The explanation at the end of +lieven lebruyn’s blog post was indeed the original motivation for the name. We were starting a new “named” lecture series as part of WAGS and wanted to name it after someone not obvious. Basque is a language not related to any other. It seemed a good idea to use that, so very few people would know the meaning of any particular word.
Then we tried all the words in WAGS, but the other three were actually very similar to the English/etc versions. The first name was chosen by vibe. Then we decided that we needed a bio for our distinguished namesake and the connection to Bourbaki presented itself for various reasons that you can guess. But we wanted a pacifist and it seemed a nice contrast to Bourbaki. So Hirune was born and we were hoping that one day she would gain prominence in the world. Finally, it happened. 🙂
June 22nd, 2014
the state of European mathematics in 1927
This map, from the Rockefeller foundation, gives us the top 3 mathematical institutes in 1927 : Goettingen, Paris and … Rome.
The pie-charts per university show that algebra was a marginal topic then (wondering how a similar map might look today).
December 1st, 2015
Did Chevalley invent the Zariski topology?
In his inaugural lecture at #ToposIHES Pierre Cartier stated (around 44m11s):
“By the way, Zariski topology, as we know it today, was not what Zariski invented. He invented a variant of that, a topology on the set of all valuation rings of a given field, which is not exactly the same thing. As for the Zariski topology, the rumour is that it was invented by Chevalley in a seminar given by Zariski, but I have no real proof.”
Do you know more about this?
Btw. the full lecture of Cartier (mostly on sheaf theory) is not on the IHES YouTube channel, but on the channel of +Laurence Honnorat.
The IHES did begin to upload videos of the remaining plenary talks here (so far, the wednesday talks are available).
The final part, starting with a reaction of Mochizuki himself on my previous IUTeich-posts.
July 7th, 2013
Mochizuki update
While i’ve been away from G+, someone emailed Mochizuki my last post on the problem i have with Frobenioids1. He was kind enough to forward me M’s reply:
“There is absolutely nothing difficult or subtle going on here (e.g., by comparison to the portion of the theory cited in the discussion preceding the statement of this “problem”!). The nontrivial result is the fact that the degree is rational, i.e., the initial portion of [FrdI], Theorem 6.4, (iii), which is a consequence of a highly nontrivial result in transcendence theory due to Lang (i.e., [FrdI], Lemma 6.5, (ii)). Once one knows this rationality, the conclusion that distinct prime numbers are not confused with one another is a formal consequence (i.e., no complicated subtle arguments!) of the fact that the ratio of the natural logarithm of any two distinct prime numbers is never rational.
Sincerely, Shinichi Mochizuki”
Being back, i’ll give it another go.
Having read the shared article below (2/5/2019 ‘the paradox of the proof’), it’s comforting to know that other people, including +Aise Johan de Jong and +Cathy O’Neil, are also frustrated by M’s opaque latest writings.
July 11th, 2013
Mochizuki’s Frobenioid reconstruction: the final bit
In “The geometry of Frobenioids 1” Mochizuki ‘dismantles’ arithmetic schemes and replaces them by huge categories called Frobenioids. Clearly, one then wants to reconstruct the schemes from these categories and we almost understood how he manages to to this, modulo the ‘problem’ that there might be auto-equivalences of $Frob(\mathbb{Z})$, the Frobenioid corresponding to $\mathbf{Spec}(\mathbb{Z})$ i.e.the collection of all prime numbers, reshuffling distinct prime numbers.
In previous posts i’ve simplified things a lot, leaving out the ‘Arakelov’ information contained in the infinite primes, and feared that this lost info might be crucial to understand the final bit. Mochizuki’s email also points in that direction.
Here’s what i hope to have learned this week:
the full Frobenioid $Frob(\mathbb{Z})$
objects of $Frob(\mathbb{Z})$ consist of pairs $(q,r)$ where $q$ is a strictly positive rational number and $r$ is a real number.
morphisms are of the form $f=(n,a,(z,s)) : (q,r) \rightarrow (q’,r’)$ (where $n$ and $z$ are strictly positive integers, $a$ a strictly positive rational number and $s$ a positive real number) subject to the relations that
$n$ is called the Frobenious degree of $f$ and $(z,s)$ the divisor of $f$.
All this may look horribly complicated until you realise that the isomorphism classes in $Frob(\mathbb{Z})$ are exactly the ‘curves’ $C(\alpha)$ consisting of all pairs $(q,r)$ such that $r-log(q)=\alpha$, and that morphisms with the same parameters as $f$ send points in $C(\alpha)$ to points in $C(\beta)$ where
\[
\beta = n.\alpha + s – log(a) \]
So, the isomorphism classes can be identified with the real numbers $\mathbb{R}$ and special linear morphisms of type $(1,a,(1,s))$ and their compositions are compatible with the order-structure and addition on $\mathbb{R}$.
Crucial is Mochizuki’s observation that $C(0)$ are precisely the ‘Frobenious-trivial’ objects in $Frob(\mathbb{Z})$ (i’ll spare you the details but it is a property on having sufficiently many nice endomorphisms).
Now, consider an auto-equivalence $E$ of $Frob(\mathbb{Z})$. It will induce a map between the isoclasses $E : \mathbb{R} \rightarrow \mathbb{R}$ which is additive and as Frob-trivs are mapped under $E$ to Frob-trivs this will map $0$ to $0$, so $E$ will be an additive group-endomorphism on $\mathbb{R}$ hence of the form $x \rightarrow r.x$ for some fixed real number $r$, and we want to show that $r=1$.
Linear irreducible morphisms are of type $(1,a,(p,0))$ where $p$ is a prime number and they map $C(\alpha)$ to $C(\alpha-log(p))$. As irreducibles are preserved under equivalence this means that for each prime $p$ there must exist a prime $q$ such that $r log(p) = log(q)$.
Now if $r$ is an irrational number, there must be at least three triples $(p_1,q_1),(p_2,q_2)$ and $(p_3,q_3)$ satisfying $r log(p_i) = log(q_i)$ but this contradicts a fairly hard result, due to Lang, that for 6 distinct primes l_1,…,l_6 there do not exist positive rational numbers $a,b$ such that
\[
log(l_1)/log(l_2) = a log(l_3)/log(l_4) = b log(l_5)/log(l_6) \]
So, $r$ must be rational and of the form $n/m$, but then $r=1$ (if not the correspondence $r.log(p) = log(q)$ gives $p^n=q^m$ contradicting unique factorisation). This then shows that under the auto-equivalence each prime $p$ (corresponding to a linear irreducible map) is send to itself.
This was the remaining bit left to show that the Frobenioid corresponding to any Galois extension of the rationals contains enough information to reconstruct from it the schemes of all rings of integers in intermediate fields.
Marabout Flash was a Franco-Belge series of do-it-yourself booklets, quite popular in the 60ties and 70ties, on almost every aspect of everyday’s life.
In just a couple of pages Szpiro describes how one can extend the structure sheaf of a fractional ideal of a ring of integers to a ‘metrized’ (or Arakelov) line-bundle on the completed prime spectrum (including the infinite places). These bundles then satisfy properties similar to those of line-bundles on smooth projective curves, including a version of the Riemann-Roch theorem and a criterium to have non-zero global sections.
These (fairly simple) results then quickly lead to proofs of the first major results in number theory such as the Hermite-Minkowski theorem and Dirichlet’s unit theorem.
“Activities surrounding IUTeich appears to be in a stage of transition from a focus on verification to a focus on dissemination.”
If only…
He further lists hypotheses as to why nobody (apart from his 3 disciples Yamashita,Saidi and Hoshi) makes a serious effort to “study the theory carefully and systematically from the beginning”:
1. it is too long (1500-2500 pages)
2. there’s lack of textbooks on anabelian geometry
3. we are obsessed with the Langlands program
4. there’s little room for generalisations
5. it may not be directly useful for our own research
But then, why should anyone make such an effort, as:
“With the exception of the handful of researchers already involved in the verification activities concerning IUTeich, every researcher in arithmetic geometry throughout the world is a complete novice with respect to the mathematics surrounding IUTeich, and hence, in particular, is simply not qualified to issue a definite judgment concerning the validity of IUTeich on the basis of a ‘deep understanding’ arising from his/her previous research achievements.”
No Mochizuki, the next phase will not be dissemination, it will be denial.
Other remarkable sentences are:
“IUTeich is ‘the correct theory’ in the sense that it leads one to doubt the existence of any sort of ‘alternative proof’, i.e. via essentially different techniques, of the ABC Conjecture.”
and:
“the status of IUTeich in the field of arithmetic geometry constitutes a sort of faithful miniature model of the status of pure mathematics in human society.”
Already looking forward to the 2015 ‘progress’ report…
October 8th, 2015
Proud to be working at a well-known university
First time I’m mentioned in “Nature”, they issue this correction:
Corrected: An earlier version of this story incorrectly located the University of Antwerp in the Netherlands. It is in Belgium. The text has been updated.
Not particularly proud of the quote they took from my blog though:
“Is it just me, or is Mochizuki really sticking up his middle finger to the mathematical community”.
December 17th, 2017
We’re heading for a bad ending
Yesterday, my feeds became congested by (Japanese) news saying that Mochizuki’s (claimed) proof of the abc-conjecture had been vetted and considered fit to be published in a respectable journal.
Today, long term supporters of M’s case began their Echternachian-retreat after finding out that Mochizuki himself is the editor in charge of that respectable journal.
Attached is Ed Frenkel’s retraction of his previous tweet. Also, Taylor Dupuy deemed the latest action a bridge too far, see here.
I have deleted my earlier tweet which I wrote being unaware that S.Mochizuki is Editor-in-Chief of the journal to which he submitted his papers. This is unfortunate. It creates the appearance of a conflict of interest & hence undermines one's confidence in the refereeing process. pic.twitter.com/LOHvpE3EYE
Peter Woit did a great job in his recent post
pinpointing a critical argument in the 500-page long papers having as its “proof” that it followed trivially from the definitions…
I’d expect any board of editors to resign in such a case.
I fear this can only end badly.
