# Riemann Hypothesis/Preliminary knowledge

## Glossary

Convergence of a series:

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

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

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

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

Dirichlet series:

A Dirichlet series is an infinite series of the form,

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

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

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

Gamma function:

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

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

Satisfying the properties,

${\displaystyle \Gamma (1)=1}$
${\displaystyle s\Gamma (s)=\Gamma (s+1)}$