IANT Meeting Notes

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
  • § 2.11: The Inverse of a Completely Multiplicative Function
    • Some Steps in the Proof of Theorem 2.17
    • More Steps in the Proof of Theorem 2.17
    • Additional Notes on Theorem 2.17
    • Proof of Theorem 2.18
  • § 2.15: Formal Power Series
    • Product of Formal Power Series
    • Commutativity of Product of Formal Power Series
    • Distributivity of Multiplication Over Addition in Formal Power Series
    • Inverse of Power Series
  • § 2.16: The Bell Series of an Arithmetical Function
    • Bell Series of $ f $ modulo $ p $
  • § 2.17: Bell Series and Dirichlet Multiplication
    • Bell Series Multiplication and Dirichlet Multiplication
    • Bell Series Multiplication and Dirichlet Multiplication: Example 1
    • Bell Series Multiplication and Dirichlet Multiplication: Example 2
    • Bell Series Multiplication and Dirichlet Multiplication: Example 3
  • § 2.18: Derivatives of Arithmetical Functions
    • Similarity of Ordinary Derivative and Special Derivative
• Chapter 3: Averages of Arithmetical Functions
  • § 3.3: Euler's Summation Formula
    • Main Idea Behind the Proof of Euler's Summation Formula
    • Sum of Integrals in the Proof of Euler's Summation Formula
    • Equation (6) in the Proof of Euler's Summation Formula
    • Using Integration by Parts in the Proof of Euler's Summation Formula
  • § 3.4: Some Elementary Asymptotic Formulas
    • Splitting Integral in Proof of Theorem 3.2
    • Euler's Constant In the Proof of Theorem 3.2 (a)
    • Some Integrals in Proof of Theorem 3.2 (b)
  • § 3.5: The Average Order of $ d(n) $
    • Rearrangement of Summations
  • § 3.11: Applications to $ \mu(n) $ and $ \Lambda(n) $
    • Final Step of Theorem 3.13
    • Illustration of Legendre's Identity
    • Illustration of the Proof of Legendre's Identity
  • § 3.12: Another Identity for the Partial Sums of a Dirichlet Product
    • Notation for Theorem 3.17
    • Theorem 3.10 as a Special Case of Theorem 3.17
    • Some Steps in Proof of Theorem 3.17
• Chapter 4: Some Elementary Theorems on the Distribution of Prime Numbers
  • § 4.2: Chebyshev's Functions $ \psi(x) $ and $ \vartheta(x) $
    • Illustration of $ \psi(x) $ and $ \vartheta(x) $
    • Some Steps in the Proof of Theorem 4.1
  • § 4.3: Relations Connecting $ \vartheta(x) $ and $ \pi(x) $
    • Some Steps in the Proof of Abel's Identity
    • From Abel's Identity to Euler's Summation Formula
    • Integration by Parts to Obtain Euler's Summation Formula From Abel's Identity
  • § 4.4: Some Equivalent Forms of the Prime Number Theorem
    • A Step in the Proof of Theorem 4.5
  • § 4.5: Inequalities for $ \pi(n) $ and $ p_n $
    • Notes on the Proof of Theorem 4.5
    • Notes on the Proof of Theorem 4.7
  • § 4.6: Shapiro's Tauberian Theorem
    • Notes on the Proof of Theorem 4.8
  • § 4.8: An Asymptotic Formula for the Partial Sums $ \sum_{p le x} (1/p) $
    • From $ \sum_{p \le x} \frac{\log p}{p} $ to $ \sum_{n \le x} a(n) \frac{\log n}{n} $
    • Applying Abel's Identity in $ \sum_{p \le x} (1/p) $
  • § 4.9: The Partial Sums of the Möbius Function
    • Applying Möbius Inversion Formula in Theorem 4.14
    • Applying Theorem 3.10 in Theorem 4.14
    • Applying Theorem 3.13 in Theorem 4.14
    • The Little-o Notation
• Chapter 5: Congruences
  • § 5.1: Definition and Basic Properties of Congruences
    • Illustration of Theorem 5.1: Reflexivity, Symmetry, and Transitivity
    • Illustration of Theorem 5.2: Addition, Subtraction, and Multiplication
    • Illustrations of Theorems 5.3 and 5.4: Cancellation Laws
    • Illustrations of Theorems 5.5-5.9
  • § 5.2: Residue Classes and Complete Residue Systems
    • Example of Residue Classes and Complete Residue System
    • Illustration of Theorem 5.11
  • § 5.3: Linear Congruences
    • Illustration of Theorem 5.12
    • Illustration of Theorem 5.13
    • Illustration of Theorem 5.14
    • Illustration of Theorem 5.14
  • § 5.4: Reduced Residue Systems and the Euler-Fermat Theorem
    • Reduced Residue System
    • Illustration of Theorem 5.16
    • Illustration of Theorem 5.17: Euler-Fermat Theorem
    • Illustration of Theorem 5.18
    • Illustration of Theorem 5.19: Little Fermat Theorem
  • § 5.5: Polynomial Congruences Modulo $ p $. Lagrange's Theorem
    • Illustration of Theorem 5.21: Lagrange's Theorem
    • Some Steps in the Proof of Theorem 5.21: Lagrange's Theorem
  • § 5.6: Applications of Lagrange's Theorem
    • Converse of Wilson's Theorem
    • Illustration of Wolstenholm's Theorem
    • Proof of Wolstenholm's Theorem
  • § 5.7: Simultaneous Linear Congruences. The Chinese Remainder Theorem
    • Solving Simultaneous Linear Congruences
    • Chinese Remainder Theorem
  • § 5.8: Applications of the Chinese Remainder Theorem
    • Illustration of Theorem 5.28
    • Illustration of Theorem 5.29
  • § 5.10: The Principle of Cross-Classification
    • Notation in the Proof of Theorem 5.31
    • Some Steps in the Proof of Theorem 5.31
    • Final Step of Euler's Totient Product Formula Example
    • Illustration of Theorem 5.32
  • § 5.11: A Decomposition Property of Reduced Residue Systems
    • Illustration of Theorem 5.33
• Chapter 6: Finite Abelian Groups and Their Characters
  • § 6.1: Definitions
    • Uniqueness of Identity Element
    • Uniqueness of Inverse Element
  • § 6.3: Elementary Properties of Groups
    • Inverse of Identity is Itself
    • Illustration of Theorems 6.1 and 6.2 With $ (\mathbb{Z}, +) $
  • § 6.4: Construction of Subgroups
    • Order of Element in Group
    • Indicator of Element in Subgroup
    • Illustration of Theorem 6.6
    • Some Steps in the Proof of Theorem 6.6
  • § 6.5: Characters of Finite Abelian Groups
    • Multiplicative Property of Characters
    • Examples of Characters
  • § 6.6: The Character Group
    • Multiplication of Characters
    • Reciprocal of Characters
    • Reciprocal and Conjugate of Characters
  • § 6.7: The Orthogonality Relations for Characters
    • Matrix of Characters
  • § 6.8: Dirichlet Characters
    • Dirichlet Characters Example
  • § 6.10: The Nonvanishing of $ L(1, \chi) $ for Real Nonprincipal $ \chi $
    • Multiplicative Property of $ A(n) $
    • Final Step of Proof of Theorem 6.20
• Chapter 7: Dirichlet's Theorem on Primes in Arithmetical Progressions
  • § 7.2: Dirichlet's Theorem for Primes of the Form $ 4n - 1 $ and $ 4n + 1
    • Illustration of Proof of Theorem 7.2
    • Some Steps in the Proofs of Theorem 7.2 and Theorem 7.3
  • § 7.4: Proof of Lemma 7.4
    • Equation (8) in the Proof of Lemma 7.4
    • Equation (9) in the Proof of Lemma 7.4
  • § 7.5: Proof of Lemma 7.5
    • Convergence of Second Sum of (11) in the Proof of Lemma 7.5
    • Equation (13) in the Proof of Lemma 7.5
    • Equation (14) in the Proof of Lemma 7.5
    • Sum in the $ O $-term in Equation (14) in the Proof of Lemma 7.5
  • § 7.6: Proof of Lemma 7.6
    • Proof of Lemma 7.6
  • § 7.8: Proof of Lemma 7.7
    • Summary of Proof of Dirichlet's Theorem
  • § 7.9: Distribution of Primes in Arithmetic Progressions
    • Illustration of Initial Steps in Proof of Theorem 7.10
• Chapter 8: Periodic Arithmetical Functions and Gauss Sums
  • § 8.1: Functions Periodic Modulo $ k $
    • Periodicity of GCD
    • Periodicity of Geometric Sum
  • § 8.2: Existence of Finite Fourier Series for Periodic Arithmetical Functions
    • Illustration of Lagrange's Interpolation Theorem
    • Summary of Lagrange's Interpolation Theorem
    • Proof of Theorem 8.4
    • Explanation of Note After Theorem 8.4
  • § 8.3: Ramanujan's Sum and Generalizations
    • Proof of Theorem 8.6
  • § 8.4: Multiplicative Properties of the Sums $ s_k(n) $
    • Last Few Steps of Theorem 8.8
    • Applying Theorem 8.8 to Ramanujan's Sum
  • § 8.5: Gauss Sums Associated with Dirichlet Characters
    • Separability of $ \chi_3 \bmod 5 $
  • § 8.8: Further Properties of Induced Moduli
    • Illustration of Theorem 8.17 with $ \chi_1 \bmod 12 $
    • Illustration of Theorem 8.17 with $ \chi_2 \bmod 12 $
    • Illustration of Theorem 8.17 with $ \chi_3 \bmod 12 $
    • Illustration of Theorem 8.17 with $ \chi_4 \bmod 12 $
  • § 8.10: Primitive Characters and Separable Gauss Sums
    • Contraposition in Proof of Theorem 8.19
    • Application of Theorem 5.33 (a) in Theorem 8.19
  • § 8.11: The Finite Fourier Series of the Dirichlet Characters
    • Summary of Proof of Theorem 8.20
  • § 8.12: Polya's Inequality for the Partial Sums of Primitive Characters
    • Equation (25) in Proof of Theorem 8.21
• Chapter 9: Quadratic Residues and the Quadratic Reciprocity Law
  • § 9.1: Quadratic Residues
    • Quadratic Residues Modulo 11
  • § 9.2: Legendre's Symbol and its Properties
    • Periodicity of Legendre's Symbol
    • Illustration of Euler's Criterion
  • § 9.3: Evaluation of $ (-1 \mid p) $ and $ (2 \mid p) $
    • Proof of Theorem 9.4
  • § 9.4: Gauss' Lemma
    • Illustration of Theorem 9.6
    • Illustration of Proof of Theorem 9.6
    • Final Step of Theorem 9.6
    • Illustration of Theorem 9.7
  • § 9.5: The Quadratic Reciprocity Law
    • Illustration of Quadratic Reciprocity Law
    • Inequalities in the Proof of Theorem 9.8
  • § 9.7: The Jacobi Symbol
    • Illustration of Jacobi Symbol
    • Illustration of Some Steps in the Proof of Theorem 9.10
    • Some Steps in the Proof of Theorem 9.11
  • § 9.8: Applications to Diophantine Equations
    • Examples values of $ p $ and $ m $
    • Useful Congruences for Theorem 9.12
  • § 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
    • An Example of Quadratic Gauss Sum
    • Relationship Between Quadratic Gauss Sums and Gauss Sums
    • Obtaining Equation (29)
    • Cases in Equation (30)
    • Note in Theorem 9.16
    • Obtaining $ g(0) = S(a, m) $
    • Computing $ g(z + 1) - g(z) $
• Chapter 10: Primitive Roots
  • § 10.1: The Exponent of a Number $ \bmod m $
    • Exponents of 2 and 3 modulo 7
  • § 10.4: The Existence of Primitive Roots $ \bmod p $ for Odd Primes $ p $
    • Illustration of Lemma 1
    • Illustration of Theorem 10.4
    • Another Illustration of Theorem 10.4
  • § 10.5: Primitive Roots and Quadratic Residues
    • Illustration of Theorem 10.5
  • § 10.6: The Existence of Primitive Roots $ \bmod p^{alpha} $
    • Illustration of Theorem 10.6
    • Binomial Expansion of $ (g + p)^{p-1} $ in Proof of Theorem 10.6
    • Raising Both Sides to the $ p $th Power in Proof of Lemma 2
  • § 10.10: The Index Calculus
    • Illustration of Index Calculus
    • Observations on Binomal Congruence Example
  • § 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
    • Absolute Value of $ n^s $
  • § 11.2: The Half-Plane of Absolute Convergence of a Dirichlet Series
    • Riemann Zeta Function
  • § 11.3: The Function Defined by a Dirichlet Series
    • Steps in Proof of Lemma 1
    • Illustration of Theorem 11.2
    • Inequality in Proof of Theorem 11.2
    • Steps in Proof of Theorem 11.3
  • § 11.4: Multiplication of Dirichlet Series
    • Summations in Proof of Theorem 11.5
    • Some Steps in Example 1
    • Some Steps in Example 2
    • Some Steps in Example 3
    • Some Steps in Example 4
    • Some Steps in Example 5
  • § 11.5: Euler Products
    • LHS and RHS
    • Illustration of Proof of Theorem 11.6
    • Some Convergence Tests
    • Examples
    • Euler Product for Principal Character
  • § 11.6: The Half-Plane of Convergence of a Dirichlet Series
    • Applying Abel's Identity in Lemma 2
    • Definite Integral in Lemma 2
    • Inequality in Lemma 2
  • § 11.7: Analytic Properties of Dirichlet Series
    • Description of Open Set
    • Description of Complex Analytic Function
    • Description of Uniform Convergence
    • Description of Compact Set
    • Cauchy's Integral Formula
    • Cauchy Criterion
  • § 11.8: Dirichlet Series with Nonnegative Coefficients
    • Obtaining Equation 16 in Proof of Theorem 11.13
    • Obtaining Final Equation in Proof of Theorem 11.13
  • § 11.9: Dirichlet Series Expressed as Exponentials of Dirichlet Series
    • Obtaining Dirichlet Product in Proof of Theorem 11.14
  • § 11.10: Mean Value Formulas for Dirichlet Series
    • Some Steps in Proof of Theorem 11.10
    • Complex Conjugates for Theorem 11.16
    • Applying Theorem 11.15 in Theorem 11.16
    • Obtaining Theorem 11.16 (b)
    • Obtaining Theorem 11.16 (c)
    • Obtaining Theorem 11.16 (d)
  • § 11.12: An Integral Formula for the Partial Sums of a Dirichlet Series
    • Contour Integral for Figure 11.4
    • Inequalities for Integrals
    • Solving Improper Integral for Proof of (20)
    • Contour Integral Around Pole for Figure 11.5
    • Final Integral in Proof of Lemma 4
    • Some Steps in Proof of Perron's Formula
• Chapter 12: The Functions $ \zeta(s) $ and $ L(s, \chi) $
  • § 12.1: Introduction
    • Hurwitz Zeta Function
    • $ L(s, \chi) $ As Linear Combination of Hurwitz Zeta Function
  • § 12.2: Properties of the Gamma Function
    • Illustration of $ \Gamma(n) $
    • Properties of $ \Gamma(s) $
  • § 12.3: Integral Representation for the Hurwitz Zeta Function
    • Inequality in Proof of Theorem 12.1
    • Change of Variable in Proof of Theorem 12.2
    • Steps Involved in Showing Convergence of Integral
  • § 12.4: A Contour Integral Representation for the Hurwitz Zeta Function
    • First Two Inequalities
    • Showing That Integrand is Bounded
    • Integral Along $ C_1 $
    • Integral Along $ C_2 $
    • Integral Along $ C_3 $
    • Adding the Integrals
    • Dividing by $ 2i $
    • Absolute Value of $ I_2(s, c) $
    • Final Step
  • § 12.5: The Analytic Continuation of the Hurwitz Zeta Function
    • Finding Residues
    • Computing $ \zeta(0) $
    • Computing $ \zeta(-1) $
  • § 12.6: Analytic Continuation of $ \zeta(s) $ and $ L(s, \chi) $
    • Final Step
  • § 12.7: Hurwitz's Formula for $ \zeta(s, a) $
    • Equality of $ F(1, s) $ and $ \zeta(s) $
    • Showing That Integrand Is Bounded
  • § 12.8: The Functional Equation for the Riemann Zeta Function
    • Legendre's Duplication Formula
    • Applying Legendre's Duplication Formula
    • Replacing the Product in (13)
  • § 12.9: A Functional Equation for the Hurwitz Zeta Function
    • Final Step in Proof of Theorem 12.8
    • Obtaining Riemann's Functional Equation
  • § 12.10: The Functional Equation for L-Functions
    • First Equation in Proof of Theorem 12.10
    • First Equation in Proof of Theorem 12.11
    • Final Steps in Proof of Theorem 12.11
  • § 12.11: Evaluation of $ \zeta(-n, a) $$
    • First Formula
    • Proof of Theorem 12.12
  • § 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
    • Obtaining First Equation in Proof of Theorem 12.21
    • Solving Integral in Proof of Theorem 12.21
    • Integration by Parts in Proof of Theorem 12.22
    • Final Step in Proof of Theorem 12.22
  • § 12.15: Inequalities for $ \lvert \zeta(s, a) \rvert $
    • Applying Equation (25) in Proof of Theorem 12.23
    • First Inequality After Applying Equation (25)
    • Second Inequality After Applying Equation (25)
• Chapter 13: Analytic Proof of Prime Number Theorem
  • § 13.2: Lemmas
    • Applying Abel's Identity in Proof of Lemma 1
    • Final Steps in Proof of Lemma 1
    • Inequality for Limit Inferior in Proof of Theorem Lemma 2
    • Summarizing Proof of Theorem 13.1
    • Gamma Function in Lemma 3
  • § 13.3: A Contour Integral Representation for $ \psi_1(x) / x^2 $
    • Applying Lemma 3 in Theorem 13.2
    • Applying Lemma 3 in Theorem 13.3
    • Final Step in Proof of Theorem 13.3
  • § 13.4: Upper Bounds for $ \zeta(s) $ and $ \zeta'(s) $ Near the Line $ \sigma = 1 $
    • Applying Theorem 12.21
    • Derivatives for Equation (18)
    • Integral in Proof of Theorem 13.4
  • § 13.5: The Nonvanishing of $ \zeta(s) $ on the Line $ \sigma = 1 $
    • Formula for $ G(s) $ in Proof of Theorem 13.5
  • § 13.6: Inequalities for $ \lvert 1/\zeta(s) \rvert $ and $ \lvert \zeta'(s) / \zeta(s) \rvert $
    • Reverse Triangle Inequality in Proof of Theorem 13.7
    • Finding $ \alpha $ in Proof of Theorem 13.7
    • Final Inequality in Proof of Theorem 13.7
  • § 13.8: Zero-Free Regions for $ \zeta(s) $
    • Applying Theorem 13.7 in Proof of Theorem 13.10
    • Applying Theorem 13.4 in Proof of Theorem 13.10
  • § 13.10: Application to the Divisor Function
    • Average Order $ d(n) $
    • Some Steps in Introduction to Section 13.10
    • Equivalent Relation in Note of Theorem 13.12
    • Illustration of $ S(n) $ in Proof of Theorem 13.12
  • § 13.11: Application to Euler's Totient
    • Expansion of $ -P(x) $ in Proof of Theorem 13.13
    • Partial Sum of $ a_p $
    • Final Step in Proof of Theorem 13.13
• Chapter 14: Partitions
  • § 14.1: Introduction
    • Goldbach Conjecture
  • § 14.3: Generating Functions for Partitions
    • Generating Function for Unrestricted Partitions
    • Generating Function for Partitions With Unequal Parts
  • § 14.4: Euler's Pentagonal-Number Theorem
    • Generating Function for Euler's Pentagonal Theorem
    • Examples of Pentagonal Numbers
    • Obtaining $ F_1 $ and $ S_1 $
    • Elaborating Steps for $ F_n - F_{n-1} $
    • Last Few Steps in Proof of Theorem 14.3
  • § 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) $
    • Illustration of Theorem 14.4
    • Proof of Theorem 14.4
  • § 14.7: An Upper Bound for $ p(n) $
    • Expansion of $ -\log(1 - x^n) $ in Proof of Theorem 14.5
    • Finding the Minimum in Proof of Theorem 14.5
    • Final Step in Proof of Theorem 14.5
    • Inequality in the Note After Theorem 14.5
    • Finding minimum for $ f(t) $ in the Note After Theorem 14.5
    • Final Step in the Note After Theorem 14.5
  • § 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
    • Obtaining Euler's Pentagonal Number Theorem from Jacobi's Identity
    • Explaining Equation (21) of Theorem 14.7
    • Replacing $ z^2 $ by $ -xz $ in Jacobi's Identity
    • Two Relations in Proof of Theorem 14.7
    • Rearrangement of Terms in Proof of Theorem 14.7
  • § 14.10: Logarithmic Differentiation of Generating Functions
    • Obtaining $ G'_A(x^m) $
    • Obtaining Equation (24) of Theorem 14.8
    • Relationship Between Divisor Sum Function and Unrestricted Partition Function
    • Illustration of Example 1
    • Elaborating Example 2
    • Elaborating Equation (25)
    • Result from Equation (25)