Riemann Hypothesis/Preliminary knowledge

GlossaryEdit

Convergence of a series:

Take some complex-valued sequence,  , and let,

 

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

  as  

Dirichlet series:

A Dirichlet series is an infinite series of the form,

 

Where   and   are complex numbers, and   is some strictly increasing sequence of nonnegative real numbers. Within this book we look at a special case. That is the case of  , a Dirichlet L-series,

 
 

Gamma function:

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

 

Satisfying the properties,