View source for Riemann Hypothesis/Preliminary knowledge

==Glossary==
{{TextBox| M=0 | W=100% | BG=#FFFFFF | clr=none |1='''Convergence of a series''':

Take some complex-valued sequence, <math>a_n</math>, and let, 

:<math>S_N = \sum_{n=1}^N a_n</math>

The sequence <math>S_N</math> may be referred to as convergent if <math>S_N</math> tends towards some finite limit, <math>L</math> as <math>N</math> becomes infinitely large. Specifically, the difference between <math>S_N</math> and <math>L</math> becomes arbitrarily small as <math>N</math> becomes arbitrarily large, ie. 

:<math>|S_N - L| \to 0</math> as <math>N \to \infty</math>

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{{TextBox| M=0 | W=100% | BG=#FFFFFF | clr=none |1='''Dirichlet series''':

A Dirichlet series is an infinite series of the form,  

:<math>\sum_{n=1}^\infty a_n\exp(-\lambda_n s)</math>

Where <math>a_n</math> and <math>s</math> are complex numbers, and <math>\lambda_n</math> is some strictly increasing sequence of nonnegative real numbers. Within this book we look at a special case. That is the case of <math>\lambda_n = \log n</math>, a Dirichlet L-series,

:<math>\sum_{n=1}^\infty a_n\exp(-\log(n) s)</math>
:<math>=\sum_{n=1}^\infty a_nn^{-s} = \sum_{n=1}^\infty \frac{a_n}{n^s}</math>
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{{TextBox| M=0 | W=100% | BG=#FFFFFF | clr=none |1='''Gamma function''':

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

:<math>s! = \Gamma(s+1) = \int_0^\infty t^{s}e^{-t}\mathrm{d}t</math>

Satisfying the properties, 

:<math>\Gamma(1) = 1</math>
:<math>s\Gamma(s) = \Gamma(s+1)</math>
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