9.1 Definition and properties of Dirichlet characters

Dirichlet characters are special functions that map integers to complex numbers, crucial for understanding prime distribution. They're like mathematical spies, revealing hidden patterns in numbers by assigning them unique complex values based on their properties.

These characters form a bridge between algebra and analysis in number theory. By studying their behavior, we can unlock secrets about primes, L-functions, and even solve tricky problems in cryptography. It's like having a secret code to decipher the mysteries of numbers.

Definition and Basic Properties

Dirichlet Characters and Their Modulus

• Dirichlet character represents a group homomorphism from multiplicative group of integers modulo n to complex numbers of unit magnitude
• Modulus n determines the periodicity of the character, defining its repetition pattern
• Characters exhibit periodic behavior repeating every n values due to their modulus
• Function χ(a) maps integers a to complex numbers, satisfying χ(a + n) = χ(a) for all integers a
• Modulus impacts character's behavior, influencing its values and properties

Multiplicative Properties of Characters

• Dirichlet characters possess multiplicative properties essential to their definition and applications
• Multiplicative function satisfies χ(ab) = χ(a)χ(b) for all coprime integers a and b
• Completely multiplicative function extends this property to all integer pairs, not just coprime ones
• Characters preserve multiplication structure when mapping from integers to complex numbers
• Multiplicativity simplifies calculations and enables powerful analytical techniques in number theory

Complex-Valued Nature and Unit Circle

• Dirichlet characters map integers to complex numbers on the unit circle
• Values of χ(a) always have magnitude 1, lying on the complex unit circle
• Character values often expressed as roots of unity ($e^{2πi/n}$)
• Complex-valued nature allows characters to capture intricate arithmetic properties
• Unit circle constraint ensures characters preserve certain algebraic structures

Special Characters

Primitive and Induced Characters

• Primitive character cannot be induced from a character of smaller modulus
• Induced characters derive from primitive ones by composition with natural projection
• Primitive characters form building blocks for all Dirichlet characters
• Process of inducing characters creates relationships between different moduli
• Identifying primitive characters crucial for understanding character structure and properties

Principal Character and Its Properties

• Principal character χ₀ maps all integers coprime to modulus to 1
• Serves as identity element in character group
• Principal character values: χ₀(a) = 1 if gcd(a,n) = 1, and χ₀(a) = 0 otherwise
• Plays special role in character theory, analogous to trivial homomorphism in group theory
• Understanding principal character essential for analyzing character sums and L-functions

Order and Cyclicity of Characters

• Order of a character defines smallest positive integer k such that χᵏ = χ₀
• Characters of order 2 called quadratic characters, having special properties
• Order relates to the multiplicative order of character values in complex plane
• Finite order of characters reflects underlying group structure
• Analyzing character orders provides insights into cyclicity and structure of character groups

Character Group

Structure and Properties of Character Groups

• Character group consists of all Dirichlet characters modulo n
• Forms an abelian group under pointwise multiplication
• Isomorphic to (Z/nZ)*, connecting characters to modular arithmetic
• Group structure allows application of group theory techniques to character analysis
• Size of character group relates to Euler's totient function φ(n)

Duality and Orthogonality Relations

• Characters exhibit duality with the group they describe
• Orthogonality relations between characters fundamental to many proofs in analytic number theory
• Sum of χ(a)χ'(a) over all a mod n equals 0 if χ ≠ χ', and φ(n) if χ = χ'
• Orthogonality enables character sum evaluations and Fourier analysis techniques
• Duality connects character theory to representation theory of finite abelian groups

Applications in Number Theory

• Character sums play crucial role in estimating prime distribution
• L-functions associated with characters generalize Riemann zeta function
• Characters used in proving quadratic reciprocity and generalizations
• Dirichlet's theorem on primes in arithmetic progressions relies heavily on character theory
• Modern cryptography employs character theory in various algorithms and protocols