## Remarks on ζ(2n+1)

December 15, 2016

[Posted on december 15, 2016.]

Having for a few days stupidly and incorrectly claimed a proof of the irrationality of all $\zeta(2n+1)$ after getting confused with a definition (sic) for which I express my deepest apologies for any confusion this may have caused, a new version of the preprint has withdrawn that claim.

The little that survives then is the formula $\displaystyle \frac{\zeta(2n+1)}{r^{2n}}=(-1)^{n+1} \int\limits_{[0;1]^n} \left (\prod_{i=1}^{n}\frac{\log(x_i)}{x_i}\right ) \log\left (1-\left(\prod_{i=1}^nx_i\right)^r \right ) dx_1\cdots dx_n$ which for $r=1$, due to its product form, I thought might have some uses that the classical $\displaystyle \zeta(k)=\int\limits_{[0;1]^k} \frac{dx_1\cdots dx_k}{1-x_1\cdots x_k}$ might not have.

Related expressions that I then obtained, namely $\displaystyle \int\limits_{[0;1]^2} \frac{\log(x)}{x}\frac{\log(y)}{y}\log(1-(xy)^1) \log(x)\log(y) (xy)^{2k+1} dxdy =$
$\displaystyle \frac{4\zeta(6)}{(2k+1)} + \frac{4\zeta(5)}{(2k+1)^2} + \frac{4\zeta(4)}{(2k+1)^3} + \frac{4\zeta(3)}{(2k+1)^4} + \frac{4\zeta(2)}{(2k+1)^5} -\frac{4}{(2k+1)^6}\sum_{n=1}^{2k+1}\frac{1}{n}-4\sum_{j=1}^{6}\left (\frac{1}{(2k+1)^j} \sum_{i=1}^{2k+1}\frac{1}{i^{7-j}}\right )$

and

$\displaystyle \int\limits_{[0;1]^2} \frac{\log(x)}{x}\frac{\log(y)}{y}\log(1-(xy)^2) \log(x)\log(y) (xy)^{2k+1} dxdy =$
$\displaystyle \frac{63\zeta(6)}{8(2k+1)} + \frac{31\zeta(5)}{4(2k+1)^2} + \frac{15\zeta(4)}{2(2k+1)^3} + \frac{7\zeta(3)}{(2k+1)^4} + \frac{6\zeta(2)}{(2k+1)^5}$
$\displaystyle +\frac{8\log(2)}{(2k+1)^6}-\frac{4}{(2k+1)^6}\sum_{i=1}^{k}\frac{1}{2i+1}-8\sum_{j=1}^{6}\left (\frac{1}{(2k+1)^j}\sum_{i=0}^{k}\frac{1}{(2i+1)^{7-j}}\right )$

do not seem to be of any help towards a proof of irrationality of at least $\zeta(5)$.  A combination of those two expressions, perhaps with other expressions such as those of Vasilyev, might allow to improve matters, but I was not able to find how.

## Some december 2016 short news

December 4, 2016

[Posted on december 4, 2016.]

Seen recently:

• the december issue of the Newletter of the EMS has appeared, with lots of interesting historical informations on the work and era of Shannon, Atiyah and Vinberg in particular
• results of the TIMSS international study have created a noticeable press coverage in France,  since they show that the abilities of french pupils in mathematics from primary school all the way to baccalauréat have dropped tremendously in the past 20 years, and place France virtually at the bottom in Europe ; as a former math teacher I’m not surprised at all…
• a gallery for middle and high school pupils dedicated to math will open next week in London
• a recent paper of Alain Connes and Farzad Fathizadeh is especially daunting “this term involves exceedingly lengthy expressions and at times involves manipulations on a few hundred thousand terms, only the final outputs of the calculations are written in this paper
• next week at CIRM a Small Group event will discuss Deformation Theory, Completed Cohomology, Leopoldt Conjecture and K-Theory
• a movie about the history of Institut Fourier, featuring among others a 95 years-old Jean-Louis Koszul
• ICM 2022 will take place either in Paris or St.Petersburg (both great choices!)
• still a few weeks left to suggest names for Prize Winners at ICM 2018 (looking at eligible candidates, it looks like it will be an especially difficult job for the Fields Medals committee this time around, with definitely more than 4 super-strong candidates)
• the first part of the Christies sale of a scientific library did very well, including £112,900 for a 1482 first edition of Euclid (more than double the lowest estimate) and a solid £266,500 for a copy of Newton’s Principia Mathematica

Orders of magnitude lower, a recent acquisition:

1892 Licence diploma of Marcel Sibuet

(a biography from Roland Brasseur’s informative

## First quick look at Semantic Scholar

November 14, 2016

[Posted on november 14, 2016]

Since the arXiv has announced today that it is teaming up with Semantic Scholar for a year (a search engine that has received some publicity lately), let’s have a first quick look.

What does it say for Jean-Pierre Serre ? Only two “influential publications” to his name, this is surely ridiculous. Wait, the earliest date of publication is set to 1989, and I can’t find a way to change it…

Let’s pick some recent authors then: Terence Tao. Now the default range is 1978-2016, and the 6th most relevant paper is “Professor Terence Tao Visit – 27 August Canterbury Statistics Open Day” by Jenifer Brown.  Not too convincing either.

One more try: Cedric Villani (to simulate someone typing with a qwerty keyboard without accents). Just one page of results. What difference an accent aigu makes. The suggestions in the toolbar only proposed some Cedrics and no Cédric.

I may have missed some obvious settings, but if not I don’t think it is very useful yet.

***

In other news:

## AMS data on the backlog of math journals

November 11, 2016

[Posted on november 11, 2016.]

Since every year in november a new batch of data regarding the backlogs of mathematics research journals is published in Notices of the AMS, let’s look at this a little. Here is the data from the past 6 years : 2015, 2014, 2013, 2012, 2011, 2010, 2009.

Now, while backlogs are one thing, perhaps the most significant column is “Median Time (in months) from Submission to Final Acceptance”. The results for a subjective selection of some of the most well-known titles are as follows (click to enlarge) :

Those median times are thus usually very steady and under 12 months, which is conforting, but with some occasional sudden marked increases for the very select journals that can reach 20 months, which can be a problem for young folks on the job market…

Are there important informations from the whole data worth mentioning beyond this? Comments welcome.

***

In other news, some items noticed recently :

• there will be a Gabberfest next june at IHÉS with an A-list of speakers (there are several anecdotes about Gabber’s aura on the web, whether on MO or blogs)
• next summer at the Newton Institute is due to take place a promising Big Proofs Programme aimed “at the challenges of bringing proof technology into mainstream mathematical practice
• the Institut Fourier turns 50
• the 5th digit in arXiv identifiers was used fo the first time last month, the counter reaching 10100
• integral calculus was performed for the first time by Leibnitz on this very day, some 341 years ago
• some people are remarkably versatile: Pierre Jalinière’s very recent PhD Thesis deals with “three independent works in cryptography, p-adic Hodge theory and Numerical analysis“!

Heart-shaped devotion by Marneejill on flickr

(which, for some reason, I’m inclined to rename as Two Americas…)

## Further october 2016 items

October 26, 2016

[Posted on october 26, 2016.]

In no particular order :

• the laureate of the ENS-Mitsubishi Heavy Industries Chair at ÉNS is Dmitry Chelkak
• some recent videos at Collège de France : the beginnings both of Claire Voisin’s course on Topologie des Variétés Algébriques, and of Marco Robalo’s Cours Peccot on Géométrie Algébrique Dérivée et Invariants de Gromov-Witten
• the october issue of Gazette des Mathématiciens is just out, with lots of great articles and information. In particular there’s an announcement of a 10 day contest in 2017 covering a broad range of topics and reserved to Masters Students in France to attract them to a research career (and for faculty to discover talented ones beyond just their exam results, I guess) : potentially a great idea depending on how it is organized
• there’s an erratum (not a very critical one luckily) for a paper published by Joan S. Birman and Hugh M. Hilden in Annals of Mathematics some 43 years ago
• an article in THE says academics are now turning down UK jobs given the Brexit situation. I do fear this is only just the beginning of indeed a massive exodus…
• hopefully Przemyslaw Chojecki’s DeepAlgebra will take off soon, the proposal seems to articulate a reasonable strategy and set of tools (I’d happily contribute if only I had more spare time…)

1652 monster door

(Saint-Malo, october 2016, Public Domain)

## Quick early october 2016 links

October 1, 2016

[Posted on october 1, 2016]

In a rush, a list of some recent items :

• Claire Voisin has won the CNRS Gold Medal, a highly prestigious award (last mathematician to win it: Alain Connes in 2004), more info here and an interview in english here
• an introductory article by Ivan Fesenko on IUT has just appeared in inference-review, and Artur Jackson has set up an IUT wiki (mostly empty for now)
• the lecture notes of the courses given by Laurent Lafforgue and by Olivia Caramello at Nottigham last summer are now available respectively here and here
• Lieven Lebruyn has started another enjoyable blog series, this time on the very recent work of Robert A. Kucharczyk & Peter Scholze
• the meeting on Open Access journals in Rennes this week was very interesting, the format allowing for lots of questions. Hopefully the beamers will be put online soon, and since it has been filmed the content of the Q&A will probably be available more widely too. [Update(oct 2): since Thierry Bouche has put his beamer on his webpage, I now feel free to mention one of the good news heard then : all Cedram journals will become Diamond Open Access in 2017 (so that will add the 3 that were not yet so). In particular Ann Fac Sci Toulouse and J. Theor Num Bordeaux are two well established and high-quality journals so that’s a very welcome move.]
• there is still time to register to the kaggle competition on OEIS data, which ends this month
• a paper by Andrej Bauer due to appear in the Bulletin of the AMS is titled Five stages of accepting constructive mathematics, it should be a very interesting read

the Burrell Collection, by Alex Livet on flickr

## Other september 2016 items

September 18, 2016

[Posted on september 18, 2016.]

Recent items spotted :

• the passing of Jean-Christophe Yoccoz at the still young age of 59 has been unanimously lamented : SMF, CNRS, AMS, a french math forum… A conference had been planned for his 60th birthday next year.
• there’s a one day event due to occur at the end of the month in Rennes whose aim is to inform local mathematicians on the current state and planned evolution of scientific publishing, in particular regarding Diamond Open Access journals apparently
• the Hot Topics workshop at MSRI next march on Galois Theory of Periods and Applications is open for registration
• Michael Atiyah & Nicholas Manton have recently released a preprint introducing a model of neutral atoms that uses compact complex algebraic surfaces
• Edward Witten has asked a question on MathOverflow regarding some group cohomology formulas introduced by condensed matter physicists
• Robert A. Kucharczyk & Peter Scholze have just posted to the arXiv a paper constructing (for some, but not all, fields of characteristic 0) a certain compact Hausdorff space whose profinite fundamental group agrees with the absolute Galois group of that field
• for those with a few millions to spare, Sotheby’s will offer for auction John F. Nash Jr’s Nobel Prize medal next months…

## Early september items

September 4, 2016

[Posted on september 4, 2016.]

An assortment of recent short news items :

Metz, a farewell. (August  2016, Public Domain).

## Who was Joseph Ser ?

August 25, 2016

In an interesting paper, Iaroslav Blagouchine has presented, among other things, the equivalence of the explicit analytic continuation of $\zeta$ due to Hasse  (in this 1930 paper) $\zeta(s)=\frac{1}{1-s}\sum_{n=0}^{\infty}\frac{1}{n+1}\sum_{k=0}^n(-1)^k \begin{pmatrix}n\\ k\end{pmatrix} (k+1)^{1-s}$  with one derived 4 years earlier by a Joseph Ser (in this paper) $\zeta(s)=\frac{1}{1-s}\sum_{n=0}^{\infty}\frac{1}{n+2}\sum_{k=0}^n(-1)^k \begin{pmatrix}n\\ k\end{pmatrix} (k+1)^{-s}$.

It may be that these could help improve the situation regarding the location of the non-trivial zeros of $\zeta$, although of course the classical Mertens approach with logs and the trigonometric identity cannot be mimicked here due to the ever growing number of terms added for each new $n$.

But who was Joseph Ser then ? There’s a short wikipedia bio saying that nothing much is known about him. There seems to be a genealogical entry for him here, with little more details beyond dates.  I couldn’t find his name among Normaliens nor Polytechniciens. He also doesn’t appear in the list of pre-1901 professeurs agrégés, and neither in the list of PhDs compiled by Hélène Gispert in her book.

As for published material, Numdam has 3 papers from him, where only the earliest one has a tiny indication: Nantes.  One can also find that a book of Ser, Les Calculs Formels des Séries de Factorielles, got a quite unfavorable review  in Bull AMS, while the same book got a more positive review in L’Enseignement Mathématique. And finally, he seems to have authored several articles in Mathesis, at least until 1953 when he was well into his seventies.

Feel free to comment if you know more about his work and career.

Caniculaire. (Metz, late august 2016, Public Domain.)

## Further august 2016 news

August 18, 2016

[Posted on august 18, 2016.]

In no particular order :

• there is a website to gather comments on the forthcoming MSC 2020 (one obvious set of candidates not yet mentionned are the various perfectoid structures–Peter Scholze had used a bunch of MSC 2010 classes, but with more than 120 citations to that paper by now it is clear new ones are needed)
• statistics papers in Inventiones are very rare, so this one must be outstanding
• Cédric Villani will give a public lecture in Mumbai tomorrow
• a student who ranked joint first at this year’s highly difficult competitive exam to enter l’X credits the book What is mathematics (2nd edition) by Courant, Robbins and Stewart, which he read while in eleventh grade (première), for giving him a passion for maths
• also joint first was Cécile Gachet, who equally ranked first at the even more difficult competitive exam to enter ÉNS Ulm (this feat was obviously saluted, e.g. on twitter)
• the MSRI program on Geometric Group Theory has just started, with in particular an introductory workshop next week
• there’s an interesting article on special values of Zeta functions in the september issue of Notices of the AMS (including $\zeta(2)$ as the volume of a moduli space)
• topically, this blog’s host has attempted to come up with something at least remotely interesting a few weeks ago, it didn’t go so well as usual, but hopefully the first section is indeed new and worthy of publication, we shall see…

Louis XVIth, the seagull, and the antiparallelogram at dawn.

Nantes, august 2016 (public domain).