Tag:Riemann

• 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}) […]

• RH and the Ishango bone

“She simply walked into the pond in Kensington Gardens Sunday morning and drowned herself in three feet of water.” This is the opening sentence of The Ishango Bone, a novel by Paul Hastings Wilson. It (re)tells the story of a young mathematician at Cambridge, Amiele, who (dis)proves the Riemann Hypothesis at the age of 26,…

• eBook ‘geometry and the absolute point’ v0.1

In preparing for next year’s ‘seminar noncommutative geometry’ I’ve converted about 30 posts to LaTeX, centering loosely around the topics students have asked me to cover : noncommutative geometry, the absolute point (aka the field with one element), and their relation to the Riemann hypothesis. The idea being to edit these posts thoroughly, add much…

• Art and the absolute point (3)

Previously, we have recalled comparisons between approaches to define a geometry over the absolute point and art-historical movements, first those due to Yuri I. Manin, subsequently some extra ones due to Javier Lopez Pena and Oliver Lorscheid. In these comparisons, the art trend appears to have been chosen more to illustrate a key feature of…

• 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…

• NeB : 7 years and now an iPad App

Exactly 7 years ago I wrote my first post. This blog wasn’t called NeB yet and it used pMachine, a then free blogging tool (later transformed into expression engine), rather than WordPress. Over the years NeB survived three hardware-upgrades of ‘the Matrix’ (the webserver hosting it), more themes than I care to remember, and a…

• 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…

• big Witt vectors for everyone (1/2)

Next time you visit your math-library, please have a look whether these books are still on the shelves : Michiel Hazewinkel‘s Formal groups and applications, William Fulton’s and Serge Lange’s Riemann-Roch algebra and Donald Knutson’s lambda-rings and the representation theory of the symmetric group. I wouldn’t be surprised if one or more of these books…

• 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…

• best of 2008 (2) : big theorems

A comment to Charles Siegel’s ‘big theorems’-series got me checking my stats.