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
13.1 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
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