Category Archives: number-theory

## Automorphic representations: a short list of books

This is a short list of books to get you started on learning automorphic representations. Before I talk about them, I will first define automorphic representation, which will take a few paragraphs. To start, we need an affine algebraic $F$-group scheme $G$ where $F$ is a number field or function field. We let $\A_F$ be […]

## Odd perfect numbers: lower bound

A perfect number is a positive integer $n$ such that $n$ is the sum of its proper divisors. For example $6 = 1 + 2 + 3$. The symbol $\sigma(n)$ is usally used for the sum of all the divisors of a positive integer $n$, so that a number is perfect if and only if […]

## Graph: number of primes containing a given digit

You can ask lots of questions about primes. After posting 50 facts about primes, I couldn't resist making another graph. In this one, the x-axis is $n$ and the y-axis is the number of primes up to $n$ that contain a given decimal digit (written in decimal, of course). I've plotted all of these on […]

## 50 Awesome facts about prime numbers

A prime is a natural number greater than one whose only factors are one and itself. I find primes pretty cool, so I made a list of 50 facts about primes: The first twenty primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, […]

## Links to Atiyah's preprints on the Riemann hypothesis

Sir Michael Atiyah's preprints are now on the internet: The Riemann Hypothesis The Fine Structure Constant The meat of the claimed proof of the Riemann hypothesis is in Atiyah's construction of the Todd map $T:\C\to \C$. It supposedly comes from the composition of two different isomorphisms $$\C\xrightarrow{t_+} C(A)\xrightarrow{t^{-1}_{-}} \C$$ of the complex field $\C$ with […]