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Analytic Number Theory Course Notes:
Math 507, Spring 2024, Boise State University
Zach Teitler
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Front Matter
1
Bertrand’s Postulate
1.1
Prime Number Theorem
1.2
Erdős’s proof of Bertrand’s postulate
1.3
Corollaries
1.3.1
Primes between consecutive squares
1.3.2
Pairs that add up to primes
1.3.3
Explicit bounds for the PNT
2
Arithmetic Functions I
2.1
Examples of arithmetic functions
2.2
Multiplicative functions
2.3
Dirichlet product
2.3.1
Basic properties of Dirichlet product
2.3.2
Dirichlet product of multiplicative functions
2.3.3
Explanation of Dirichlet product
2.4
Möbius inversion
2.4.1
Möbius function
2.4.2
Möbius function and Dirichlet product
3
Floor Function
3.1
Elementary properties of the floor function
3.2
Hermite’s Identity
3.3
Applications of Hermite’s Identity
4
Summation Formulas
4.1
Big O Notation And Friends
4.1.1
Big O Notation
4.1.2
Little o Notation
4.2
Sums of Monotone Functions
4.3
Summation by Parts
4.3.1
Integration by parts
4.3.2
Summation by parts
4.4
Riemann-Stieltjes Integration
4.4.1
Riemann integral
4.4.2
Riemann-Stieltjes integration
4.4.3
Step functions
4.5
Abel Summation
4.6
Euler Summation
5
Arithmetic Functions II
5.1
Average Order of Growth
5.1.1
Average order of number of divisors
5.1.2
Average order of sum of divisors
5.1.3
Average order of Euler totient
5.1.4
Extensions
5.2
Dirichlet’s Hyperbola Method
5.2.1
Euler-Mascheroni constant
5.2.2
Hyperbola method
6
Elementary Results on the Distribution of Primes
6.1
Elementary results on distribution of primes
6.2
Chebyshev’s Estimates
6.3
Mertens’ Theorems
7
Dirichlet’s Theorem
7.1
Overview of Proof of Dirichlet’s Theorem
7.1.1
Repeating the proof of Mertens’s theorems
7.1.2
Complex roots of unity
7.2
Characters of finite abelian groups
7.2.1
Definition
7.2.2
Sums of character values
7.3
Orthogonality relations
7.4
Dirichlet characters
7.5
Dirichlet L-functions
7.6
Proof of Dirichlet’s theorem, part 1
7.6.1
Going from summing over primes to summing over an arithmetic progression
7.6.2
Summing over all positive integers by using the orthogonality relations of Dirichlet characters
7.6.3
Each nonprincipal character contributes a bounded amount
7.7
Proof of Dirichlet’s theorem, part 2
7.7.1
Real nonprincipal characters
7.7.2
Nonreal nonprincipal characters
7.7.3
Conclusion: Proof of Dirichlet’s theorem
Backmatter
References
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This book was authored in PreTeXt.
