## Prime Obsession

Read Prime Obsession over the last five days (I got a hold of it just in time for vacation).  It is a really nicely written book on the history and math behind the Riemann hypothesis.  The areas of math involved here are number theory (prime numbers), and complex analysis.  The Euler product (shown on the cover) is the beginning of the bridge between the two fields.

Mathematicians are interested in the number of primes less than a given integer $N$ and call this $\pi(N)$.   While the function $\pi(N)$ can be calculated for any given $N$, there is no closed form (simple) solution for it.  Approximations have been proposed, the first being

$\pi(N) \sim\frac{N}{\log(N)}$

and the second being

$\pi(x) \sim Li(x) = \int \limits_0^x \frac{1}{\log(t)} dt$

and the closest approximation coming in the last chapter.

Meanwhile, also of interest are the convergent infinite series of fractions, given by the Riemann zeta function $\zeta(s)$.  This function is defined as

$\zeta(s) = 1 + \frac{1}{2^s} + \frac{1}{3^s} + \frac{1}{4^s} + ...$

The first solution to this I encountered in college for $s=2$ which is amazingly

$1 + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + ... = \frac{\pi^2}{6}$

Euler solved this generally for all $N$ even, (and they are all powers of $\pi$ to the $N$).  However, all $N$ odd are unsolved, and it was not until 1978 that $N=3$ was even proved to be irrational!  This summed series can be converted with regular algebra (and some insight using something like the sieve of Eratosthenes) to a product series.  This is the famous Euler product

$\sum \limits_n n^{-s} = \prod \limits_p (1-p^{-s})^{-1}$

where $n$ are all the integers and $p$ stands for all prime numbers.  (The use of $s$ is a convention started by Riemann, and not disturbed by mathematicians over the centuries, it could just as well be $x$).   The Riemann zeta function is continued analytically into the complex plane and gives rise to the Riemann hypothesis,

The Riemann hypothesis – all non-trivial zeros of the zeta function have real part one-half.

(now the zeta function in the complex plane is linked with primes!)

Random notes

In the arithmetic of congruences, $8+9 \equiv 5 (\mod 12)$.

I thought Derbyshire’s comment on thought process interesting “mathematical thinking is a deeply unnatural way of thinking, and is probably responsible for repelling so many people” – pg. 89

A conditionally convergent series sums to different limits depending on the order of summation, as opposed to an absolutely convergent series.

$\zeta(1-s) = 2^{1-s} \pi^{-s} \sin \left( \frac{1-s}{2}\pi\right) (s-1)! \zeta(s)$

${ \cal P } \in { \cal N }$ the primes are a subset of natural numbers.

${\cal A} \in {\cal C}$ the algebraic numbers are zeros of some polynomial on ${\cal Z}$.

Non-algebraic irrationals are transcendental.

There are infinitely many rationals, and infinitely many irrationals, but these are different infinities and the irrational’s infinity is larger.

Littlewood in 1914 proved $Li(x)-\pi(x)$ changes from – to + and back infinitely many times, violating the “upper-bound” of $Li(x)$.

Skewes’ number gives that it crosses at least once by $e^{e^{e^{79}}}$ (one of the largest numbers to occur naturally in a math proof!)

The Psychology of Invention in the Mathematical Field – Hademard 1945

The Möbius function $\mu(n)$ has a domain of the natural numbers and is defined by

$\mu(1) = \begin{cases} 1, & \mbox{if } n=1 \\ 0, & \mbox{if } n \mbox{ has a square factor} \\ -1, & \mbox{if } n \mbox{ is a prime, or the product of and odd number of primes} \\ +1, & \mbox{if } n \mbox{ is a product of an even number of primes} \end{cases}$

and is used in the inversion

$\frac{1}{\zeta(s)} = \sum \limits_n \frac{\mu(n)}{n^s}$

The Mertens’s function is defined by $M(k) \equiv \sum \limits_i^k \mu(i)$

The Riemann hypothesis is equivalent to $M(k) = O(k^{1/2 + \epsilon})$ for every $\epsilon$, no matter how small.

“We must know, we shall know.” – Hilbert 1930.

A ring is a group supporting +, -, x, but not necessarily division.  A field is a ring supporting division as well.  ${\cal Q, R, C}$ are all fields, ${\cal N, Z}$ are not.  I.e. all numbers of the form $a+b\sqrt{2}$ where $a,b\in {\cal Q}$ is a field that inclues all rationals (${\cal Q}$) and some irrationals.  Fields can be finite, and can be contructed from any prime number p, and for powers of primes.  E.g. $F_2, F_4, F_6, ... F_3, F_9, F_{27}, ... F_5, F_{25}, F_{125}, ...$

Rings are named like ${\cal Z}/4{\cal Z}$, and ${\cal Z}/4{\cal Z}$ is $\neq F_4$.

The characteristic of a field tells you how many times you have to add one to itself to get 0.

“The great importance of matrices is that they can be used to represent, to quantify, certain deeper and more fundamental things.   They are operators.  … That is why the characteristic polynomial, the eigenvalues, and the trace are such key concepts, they are properties of the underlying operator, not just of the matrix.  an operator can be represented by many matrices.”

The family of NxN matrices is “the general linear group for N” or $GL_N$.

All the eigenvalues of a Hermitian matrix are real.  so, all the coeffients of the characteristic polynomial are real.

The Hilbert-Pólya conjecture – the non-trivial zeros of the Riemann zeta function correspond to the eigenvalues of some Hermitian operator.

The Gaussian Unitary Ensemble (GUE) is a set of Gaussian-random Hermitian matrices.  Eigenvalues of these show unusual spacings described by pair-correlation function with a characteristic ratio called its form factor.  Evidence for this is that the zeta zeros and eigenvalues don’t look random, they look like each other, and both show a repulsion effect.  See 1973 – Hugh Montgomery – AMS – spacing of zeta zeros.  The Montgomery-Odlyzko Law – distributions are statistically identical.

Riemann (1859) defines

$J(x) = \pi(x) + \frac{1}{2}\pi(x^{1/2}) +\frac{1}{3}\pi(x^{1/3}) + ...$

which can be inverted to $\pi(x)$ using the Möbius $\mu$ function,

$\pi(x) = \sum \limits_n \frac{\mu(n)}{n} J(x^{\frac{1}{n}})$

This is called the Möbius inversion.

$\frac{1}{s}\log \zeta(s) = \int \limits_0^\infty J(x) x^{-s-1} dx$

In chaotic systems perturbation theory cannot hold; small changes in initial conditions lead to large effects.

The Hamilton operator encodes the systems energy.  Eigenvalues correspond to energy levels and eigenvectors to the system’s state.

The p-adic numbers are mathematical objects, created in 1897 by Kurt Hensel.  There is one field of p-adic numbers for any prime.  It is built of rings of size $p, p^2, p^3, ...$  The field from 7, ${\cal Q}_7$, is built from ${\cal Z}/7{\cal Z}$, ${\cal Z}/49{\cal Z}$, ${\cal Z}/343{\cal Z}$, …   The field ${\cal Q}_p$, like ${\cal R}$, can be used to complete ${\cal Q}$.  An adele is built from ${\cal Q}_2$, ${\cal Q}_3$, ${\cal Q}_5$, … and ${\cal R}$, and so is a class of super-numbers that are, each one, embedded with all the primes.   Alain Connes built an adele to host his Riemann operator.

If the Riemann hypothesis is true (RH), then evaluation of the error term is possible, via

$J(x) = Li(x) - \sum \limits_p Li(x^p) - \log(2) + \int \limits_x^\infty \frac{dt}{t(t^2-1) \log(t)}$