This page contains an archive of scribbling boards from our book club meetings on is Apostol, Introduction to Analytic Number Theory.

Note that this set of boards is not meant to be a systematic exposition of analytic number theory. Instead this is just a collection of examples that illustrate some of the theorems in the reference textbook and intermediate steps that are not explicitly expressed in the book. These boards were used to aid the discussions during our book club meetings. As a result, the content of these boards is informal in nature and is not intended to be a substitute for the book or the actual book club meetings.

If you find any mistakes in the content of the board files, please create a new issue or send a pull request.

**Chapter 2: Arithmetical Functions and Dirichlet Multiplication****Chapter 3: Averages of Arithmetical Functions****Chapter 4: Some Elementary Theorems on the Distribution of Prime Numbers**- § 4.2: Chebyshev's Functions $ \psi(x) $ and $ \vartheta(x) $
- § 4.3: Relations Connecting $ \vartheta(x) $ and $ \pi(x) $
- § 4.4: Some Equivalent Forms of the Prime Number Theorem
- § 4.5: Inequalities for $ \pi(n) $ and $ p_n $
- § 4.6: Shapiro's Tauberian Theorem
- § 4.8: An Asymptotic Formula for the Partial Sums $ \sum_{p le x} (1/p) $
- § 4.9: The Partial Sums of the Möbius Function

**Chapter 5: Congruences**- § 5.1: Definition and Basic Properties of Congruences
- § 5.2: Residue Classes and Complete Residue Systems
- § 5.3: Linear Congruences
- § 5.4: Reduced Residue Systems and the Euler-Fermat Theorem
- § 5.5: Polynomial Congruences Modulo $ p $. Lagrange's Theorem
- § 5.6: Applications of Lagrange's Theorem
- § 5.7: Simultaneous Linear Congruences. The Chinese Remainder Theorem
- § 5.8: Applications of the Chinese Remainder Theorem
- § 5.10: The Principle of Cross-Classification
- § 5.11: A Decomposition Property of Reduced Residue Systems

**Chapter 6: Finite Abelian Groups and Their Characters**- § 6.1: Definitions
- § 6.3: Elementary Properties of Groups
- § 6.4: Construction of Subgroups
- § 6.5: Characters of Finite Abelian Groups
- § 6.6: The Character Group
- § 6.7: The Orthogonality Relations for Characters
- § 6.8: Dirichlet Characters
- § 6.10: The Nonvanishing of $ L(1, \chi) $ for Real Nonprincipal $ \chi $

**Chapter 7: Dirichlet's Theorem on Primes in Arithmetical Progressions****Chapter 8: Periodic Arithmetical Functions and Gauss Sums**- § 8.1: Functions Periodic Modulo $ k $
- § 8.2: Existence of Finite Fourier Series for Periodic Arithmetical Functions
- § 8.3: Ramanujan's Sum and Generalizations
- § 8.4: Multiplicative Properties of the Sums $ s_k(n) $
- § 8.5: Gauss Sums Associated with Dirichlet Characters
- § 8.8: Further Properties of Induced Moduli
- § 8.10: Primitive Characters and Separable Gauss Sums
- § 8.11: The Finite Fourier Series of the Dirichlet Characters
- § 8.12: Polya's Inequality for the Partial Sums of Primitive Characters

**Chapter 9: Quadratic Residues and the Quadratic Reciprocity Law**- § 9.1: Quadratic Residues
- § 9.2: Legendre's Symbol and its Properties
- § 9.3: Evaluation of $ (-1 \mid p) $ and $ (2 \mid p) $
- § 9.4: Gauss' Lemma
- § 9.5: The Quadratic Reciprocity Law
- § 9.7: The Jacobi Symbol
- § 9.8: Applications to Diophantine Equations
- § 9.9: Gauss Sums and the Quadratic Reciprocity Law
- First Equation of Theorem 9.13
- Unique $ t \bmod p $ in the Proof of Theorem 9.13
- Solving the Geometric Sum in the Proof of Theorem 9.13
- Final Step of Theorem 9.13
- Obtaining Equation 24 in Proof of Theorem 9.15
- Definition of $ G(n, \chi) $ Used in Proof of Theorem 9.15
- Final Steps of Proof of Theorem 9.15

- § 9.10: The Reciprocity Law for Quadratic Gauss Sums

**Chapter 10: Primitive Roots**- § 10.1: The Exponent of a Number $ \bmod m $
- § 10.4: The Existence of Primitive Roots $ \bmod p $ for Odd Primes $ p $
- § 10.5: Primitive Roots and Quadratic Residues
- § 10.6: The Existence of Primitive Roots $ \bmod p^{alpha} $
- § 10.10: The Index Calculus
- § 10.11: Primitive Roots and Dirichlet Characters
- Periodicity of $ \chi_h $
- Illustration of Equation (21) With Modulus 3
- Illustration of Equation (21) With Modulus 5
- Illustration of Equation (21) With Modulus 2
- Illustration of Equation (21) With Modulus 4
- Illustration of Theorem 10.11 With Modulus 8
- Illustration of Theorem 10.11 With Modulus 16
- Binomial Expansion in Proof of Theorem 10.11
- Dirichlet Characters Modulo 15

**Chapter 11: Dirichlet Series and Euler Products**- § 11.1: Introduction
- § 11.2: The Half-Plane of Absolute Convergence of a Dirichlet Series
- § 11.3: The Function Defined by a Dirichlet Series
- § 11.4: Multiplication of Dirichlet Series
- § 11.5: Euler Products
- § 11.6: The Half-Plane of Convergence of a Dirichlet Series
- § 11.7: Analytic Properties of Dirichlet Series
- § 11.8: Dirichlet Series with Nonnegative Coefficients
- § 11.9: Dirichlet Series Expressed as Exponentials of Dirichlet Series
- § 11.10: Mean Value Formulas for Dirichlet Series
- § 11.12: An Integral Formula for the Partial Sums of a Dirichlet Series

**Chapter 12: The Functions $ \zeta(s) $ and $ L(s, \chi) $**- § 12.1: Introduction
- § 12.2: Properties of the Gamma Function
- § 12.3: Integral Representation for the Hurwitz Zeta Function
- § 12.4: A Contour Integral Representation for the Hurwitz Zeta Function
- § 12.5: The Analytic Continuation of the Hurwitz Zeta Function
- § 12.6: Analytic Continuation of $ \zeta(s) $ and $ L(s, \chi) $
- § 12.7: Hurwitz's Formula for $ \zeta(s, a) $
- § 12.8: The Functional Equation for the Riemann Zeta Function
- § 12.9: A Functional Equation for the Hurwitz Zeta Function
- § 12.10: The Functional Equation for L-Functions
- § 12.11: Evaluation of $ \zeta(-n, a) $$
- § 12.12: Properties of Bernoulli Numbers and Bernoulli Polynomials
- Initial Identity in Proof of Theorem 12.14
- Obtaining Equation (18) in Proof of Theorem 12.14
- Obtaining Equation (19) in Proof of Theorem 12.14
- Proof of Theorem 12.15
- Computing $ B_0 $
- Computing Bernoulli Numbers
- Computing Polynomials $ B_n(x) $
- Symbolic Representations of Theorems 12.12 and 12.15
- Computing $ \zeta(0) $ and $ \zeta(-1) $ Again
- Proof of Theorem 12.16
- Showing Even Function in Note for Theorem 12.16
- Showing $ B_{2n + 1} = 0 $ in Note for Theorem 12.16
- Asymptotic Relation in Theorem 12.18
- Asymptotic Relation in Note for Theorem 12.18
- Applying Stirling's Formula to Theorem 12.18
- Obtaining $ B_n(x) $ in Theorem 12.19
- Obtaining $ B_{2n}(x) $ in Theorem 12.19
- Obtaining $ B_{2n + 1}(x) $ in Theorem 12.19

- § 12.14: Approximation of $ \zeta(s, a) $ by Finite Sums
- § 12.15: Inequalities for $ \lvert \zeta(s, a) \rvert $

**Chapter 13: Analytic Proof of Prime Number Theorem**- § 13.2: Lemmas
- § 13.3: A Contour Integral Representation for $ \psi_1(x) / x^2 $
- § 13.4: Upper Bounds for $ \zeta(s) $ and $ \zeta'(s) $ Near the Line $ \sigma = 1 $
- § 13.5: The Nonvanishing of $ \zeta(s) $ on the Line $ \sigma = 1 $
- § 13.6: Inequalities for $ \lvert 1/\zeta(s) \rvert $ and $ \lvert \zeta'(s) / \zeta(s) \rvert $
- § 13.8: Zero-Free Regions for $ \zeta(s) $
- § 13.10: Application to the Divisor Function
- § 13.11: Application to Euler's Totient

**Chapter 14: Partitions**- § 14.1: Introduction
- § 14.3: Generating Functions for Partitions
- § 14.4: Euler's Pentagonal-Number Theorem
- § 14.5: Combinatorial Proof of Euler's Pentagonal-Number Theorem
- Illustration of Case 1 $ (b < s) $
- Illustration of Case 2.1 $ (b = s) $ (no intersection)
- Illustration of Case 2.2 $ (b = s) $ (intersection)
- Illustration of Case 3.1 $ (b > s) $ (no intersection)
- Illustration of Case 3.2 $ (b = s + 1) $ (intersection)
- Illustration of Case 3.3 $ (b > s + 1) $ (intersection)
- Summary of All Cases

- § 14.6: Euler's Recursion Formula for $ p(n) $
- § 14.7: An Upper Bound for $ p(n) $
- § 14.8: Jacobi's Triple Product Identity
- Expansion of Equation (13)
- Obtaining the Functional Equation for $ F(z) $
- Obtaining the Recursion Formula for Coefficients
- Applying the Recursion Formula for Coefficients for Positive $ m $
- Applying the Recursion Formula for Coefficients for Negative $ m $
- Obtaining Equation (20)
- Obtaining Product Formula for $ G_x(e^{\pi i / 4}) $
- Final Equation for $ a_0(x) $

- § 14.9: Consequences of Jacobi's Identity
- § 14.10: Logarithmic Differentiation of Generating Functions