1a) For info on Dirichlet's Theorem check out http://www.jwanless.freeserve.co.uk (my site)
1b) Dirichlet's series are used to produce generating functions of arithmetical functions - 
 see e.g. Hardy & Wright "An Introduction to the Theory of Numbers" 5th ed. pp.244-250
1c) Dirichlet's problem refers to approximation of irrationals by rationals - 
 see e.g. Hardy & Wright pp.154-158
2) As regards Mathematica your best bet is probably trawling the net (e.g. from http://www.alta-vista.com
 searching on +dirichlet +mathematica) or looking at the resources and documentation that came with
 Mathematica, - if you don't have Mathematica yet you can get it from http://www.wolfram.com