Riemann Hypothesis/Preliminary knowledge

Glossary

Convergence of a series:

Take some complex-valued sequence, $a_{n}$, and let,

$S_{N}=\sum _{n=1}^{N}a_{n}$

The sequence $S_{N}$ may be referred to as convergent if $S_{N}$ tends towards some finite limit, $L$ as $N$ becomes infinitely large. Specifically, the difference between $S_{N}$ and $L$ becomes arbitrarily small as $N$ becomes arbitrarily large, ie.

$|S_{N}-L|\to 0$ as $N\to \infty$

Dirichlet series:

A Dirichlet series is an infinite series of the form,

$\sum _{n=1}^{\infty }a_{n}\exp(-\lambda _{n}s)$

Where $a_{n}$ and $s$ are complex numbers, and $\lambda _{n}$ is some strictly increasing sequence of nonnegative real numbers. Within this book we look at a special case. That is the case of $\lambda _{n}=\log n$, a Dirichlet L-series,

$\sum _{n=1}^{\infty }a_{n}\exp(-\log(n)s)$

$=\sum _{n=1}^{\infty }a_{n}n^{-s}=\sum _{n=1}^{\infty }{\frac {a_{n}}{n^{s}}}$

Gamma function:

The Gamma function extends the factorial to all complex numbers bar non-positive integers. It is defined by the improper integral,

$s!=\Gamma (s+1)=\int _{0}^{\infty }t^{s}e^{-t}\mathrm {d} t$

Satisfying the properties,

$\Gamma (1)=1$

$s\Gamma (s)=\Gamma (s+1)$