1 Introduction 1.1 Acknowledgements 2 Some Basics 3 Dirichlet Characters 3.1 The Orthogonality Relation of Dirichlet Characters 3.2 An Identity Involving Characters 4 L-Series 5 The Gamma Function 6 The Riemann Zeta-Function 6.1 Analytic Continuation of the Riemann zeta-function 6.2 The Riemann Hypothesis 7 The Functional Equation of L(s,x) 7.1 Gauss sums 7.2 The Functional Equation when X(-1) =1 7.3 The Functional Equation when X(-1) =-1 8 The Poisson Summation Formula 8.1 The Functional Equation for the Theta Function 9 Siegel Zeros 10 Dirichlet's Theorem on Primes in Arithmetic Progressions 10.1 An Important Result 10.2 The Proof of Dirichlet's Theorem 11 The Prime Number Theorem for Arithmetic Progressions 12 The yon Mangoldt Function 13 An Application of Analytic Number Theory: The Negative Pell Equation 13.l Introduction 13.2 Strategy for Proving the Main Uniformity Assumption 14 The Maclaurin-Cauchy Integral Formula 15 An Important Lemma 16 Proof of The Main Uniformity Assumption 17 Further Reading On The Negative Pell Equation ��݋��ӛ