Tag: modular

  • the Riemann hypothesis and Psi

    Last time we revisited Robin’s theorem saying that 5040 being the largest counterexample to the bound \[ \frac{\sigma(n)}{n~log(log(n))} < e^{\gamma} = 1.78107... \] is equivalent to the Riemann hypothesis. There’s an industry of similar results using other arithmetic functions. Today, we’ll focus on Dedekind’s Psi function \[ \Psi(n) = n \prod_{p | n}(1 + \frac{1}{p}) […]

  • Monsters and Moonshine : a booklet

    I’ve LaTeXed $48=2 \times 24$ posts into a 114 page booklet Monsters and Moonshine for you to download. The $24$ ‘Monsters’ posts are (mostly) about finite simple (sporadic) groups : we start with the Scottish solids (hoax?), move on to the 14-15 game groupoid and a new Conway $M_{13}$-sliding game which uses the sporadic Mathieu…

  • the Reddit (after)effect

    Sunday january 2nd around 18hr NeB-stats went crazy. Referrals clarified that the post ‘What is the knot associated to a prime?’ was picked up at Reddit/math and remained nr.1 for about a day. Now, the dust has settled, so let’s learn from the experience. A Reddit-mention is to a blog what doping is to a…

  • So, who did discover the Leech lattice?

    For the better part of the 30ties, Ernst Witt (1) did hang out with the rest of the ‘Noetherknaben’, the group of young mathematicians around Emmy Noether (3) in Gottingen. In 1934 Witt became Helmut Hasse‘s assistent in Gottingen, where he qualified as a university lecturer in 1936. By 1938 he has made enough of…

  • Who discovered the Leech lattice?

    The Leech lattice was, according to wikipedia, ‘originally discovered by Ernst Witt in 1940, but he did not publish his discovery’ and it ‘was later re-discovered in 1965 by John Leech’. However, there is very little evidence to support this claim. The facts What is certain is that John Leech discovered in 1965 an amazingly…

  • Langlands versus Connes

    This is a belated response to a Math-Overflow exchange between Thomas Riepe and Chandan Singh Dalawat asking for a possible connection between Connes’ noncommutative geometry approach to the Riemann hypothesis and the Langlands program. Here’s the punchline : a large chunk of the Connes-Marcolli book Noncommutative Geometry, Quantum Fields and Motives can be read as…

  • Seriously now, where was the Bourbaki wedding?

    A few days before Halloween, Norbert Dufourcq (who died december 17th 1990…), sent me a comment, containing lots of useful information, hinting I did get it wrong about the church of the Bourbali wedding in the previous post. Norbert Dufourcq, an organist and student of Andre Machall, the organist-in-charge at the Saint-Germain-des-Prés church in 1939,…

  • Bourbakism & the queen bee syndrome

    Probably the smartest move I’ve made after entering math-school was to fall in love with a feminist. Yeah well, perhaps I’ll expand a bit on this sentence another time. For now, suffice it to say that I did pick up a few words in the process, among them : the queen bee syndrome : women…

  • E(8) from moonshine groups

    Are the valencies of the 171 moonshine groups are compatible, that is, can one construct a (disconnected) graph on the 171 vertices such that in every vertex (determined by a moonshine group G) the vertex-valency coincides with the valency of the corresponding group? Duncan describes a subset of 9 moonshine groups for which the valencies…

  • looking for the moonshine picture

    We have seen that Conway’s big picture helps us to determine all arithmetic subgroups of $PSL_2(\mathbb{R}) $ commensurable with the modular group $PSL_2(\mathbb{Z}) $, including all groups of monstrous moonshine. As there are exactly 171 such moonshine groups, they are determined by a finite subgraph of Conway’s picture and we call the minimal such subgraph…