Archive for December, 2016

Some december 2016 items

December 31, 2016

[Posted on december 31, 2016.]

Some items spotted recently:

  • the next séminaire Bourbaki on saturday january 14 will feature an all-star list of speakers, and should be live on youtube
  • the list of the top 10 journals ranked by MCQ for 2015 on mathscinet had Cambridge Journal of Mathematics in 4th place, impressive for a journal launched only in 2013
  • the famous paper of Maryna S. Viazovska on sphere packings in dimension 8 has been listed in the ‘to appear’ section of Annals of Mathematics
  • Alain Connes will give the first lecture of his annual course at Collège de France next week
  • Ivan Fesenko will give next month a Colloquium style talk in Cambridge on Shinichi Mochizuki’s IUT Theory
  • major blogs have recently advertised the fine AMS Open Math Notes website, and the reader may like to know that another great site is the math section of Cours En Ligne (hosted by CCSD), which contains a fair amount of notes in english beyond the ones in french
  • two new programs are starting at MSRI next month: Analytic Number Theory lead by Terence Tao and Harmonic Analysis lead by Michael Christ and Michael Lacey, while there will be a graduate summer school in june/july on Soergel Bimodules lead by Benjamin Elias and Geordie Williamson and another one in july/august on  Automorphic Forms and the Langlands Program lead by Kevin Buzzard
  • Jean-Yves Girard has recently released a new paper which is the third part of a series on transcendental syntax (at the boundary between logic and philosopy), and he also published a less technical book on this topic last september
  • some New Year wishes from the President of the EMS (containing a shocking piece of information: “numerous colleagues registered for the Berlin congress but did not then pay, causing a financial headache for the organizers“)
  • just a few days left to register a candidacy to become CNRS researcher
  • a new website for biographies of mathematicians (as announced in Notices of the AMS)

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Amphithéâtre, by Patrick Janicek on flickr

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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

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Orders of magnitude lower, a recent acquisition:

1892 Licence diploma of Marcel Sibuet

(a biography from Roland Brasseur’s informative

dictionnaire des professeurs de mathématiques spéciales)