
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 1415 game groupoid and a new Conway $M_{13}$sliding game which uses the sporadic Mathieu…

Langlands versus Connes
This is a belated response to a MathOverflow 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 ConnesMarcolli book Noncommutative Geometry, Quantum Fields and Motives can be read as…

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 vertexvalency 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…