13.1 Multiplicative functions and their properties

Multiplicative functions are the building blocks of number theory, satisfying f(ab) = f(a)f(b) for coprime integers. They include the Möbius function, Liouville function, and completely multiplicative functions, which extend this property to all integers.

These functions are crucial in analytic number theory, connecting to prime factorization and the Riemann hypothesis. They interrelate through operations like Dirichlet convolution and Dirichlet series, providing powerful tools for studying number-theoretic problems.

Multiplicative Functions

Fundamental Multiplicative Functions

• Multiplicative function satisfies $f(ab) = f(a)f(b)$ for all coprime integers a and b
• Completely multiplicative function extends this property to all integers, not just coprime ones
• Möbius function μ(n) assigns values based on prime factorization of n:
  • μ(1) = 1
  • μ(n) = (-1)^k if n is a product of k distinct primes
  • μ(n) = 0 if n has a squared prime factor
• Liouville function λ(n) defined as (-1)^Ω(n), where Ω(n) counts total number of prime factors of n (with multiplicity)

Properties and Applications

• Multiplicative functions preserve multiplication structure of integers
• Completely multiplicative functions form a subset of multiplicative functions
• Möbius function plays crucial role in number theory, particularly in Möbius inversion formula
• Liouville function relates to Riemann hypothesis through its summatory function

Arithmetic Functions

Common Arithmetic Functions

• Divisor function d(n) counts number of positive divisors of n, includes 1 and n itself
• Mangoldt function Λ(n) defined as:
  • Λ(n) = log p if n is a power of prime p
  • Λ(n) = 0 otherwise
• Euler's totient function φ(n) counts number of integers up to n that are coprime to n

Properties and Relationships

• Divisor function relates to prime factorization: if n = p1^a1 * p2^a2 * ... * pk^ak, then d(n) = (a1+1)(a2+1)...(ak+1)
• Mangoldt function connects to prime number theorem and distribution of primes
• Euler's totient function satisfies φ(n) = n * ∏(1 - 1/p) for all prime factors p of n
• These functions interrelate through various identities and formulas in number theory

Function Operations

Dirichlet Convolution

• Dirichlet convolution (f * g)(n) = Σd|n f(d)g(n/d) combines two arithmetic functions
• Results in new arithmetic function
• Preserves multiplicativity: if f and g are multiplicative, so is f * g
• Identity element for Dirichlet convolution I(n) = 1 if n = 1, and 0 otherwise

Dirichlet Series

• Dirichlet series representation of arithmetic function f(n): F(s) = Σn=1 to ∞ f(n)/n^s
• Provides analytic tool for studying arithmetic functions
• Allows use of complex analysis techniques in number theory
• Multiplication of Dirichlet series corresponds to Dirichlet convolution of their coefficient functions