2.3 Additive functions and multiplicative functions

Additive and multiplicative functions are key players in number theory. They satisfy special properties when applied to coprime numbers, allowing us to break down complex problems into simpler parts.

These functions, like the Möbius and Euler totient functions, help solve tricky number theory puzzles. They're crucial for understanding prime numbers, divisors, and other fundamental concepts in additive number theory.

Additive vs Multiplicative Functions

Definitions and Properties

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• Additive functions are arithmetic functions $f(n)$ that satisfy $f(ab) = f(a) + f(b)$ whenever $gcd(a,b) = 1$
  • The sum of two additive functions is also additive
• Multiplicative functions are arithmetic functions $f(n)$ that satisfy $f(ab) = f(a)f(b)$ whenever $gcd(a,b) = 1$
  • The product of two multiplicative functions is also multiplicative

Examples

• Examples of additive functions include:
  • The sum of divisors function $\sigma(n)$
  • The totient function $\phi(n)$
• Examples of multiplicative functions include:
  • The Möbius function $\mu(n)$
  • The Euler totient function $\phi(n)$
  • The divisor function $d(n)$

Properties of Arithmetic Functions

Common Arithmetic Functions

• Arithmetic functions are real or complex-valued functions defined on the set of positive integers
• Common arithmetic functions include:
  • The Möbius function $\mu(n)$
    • Defined as: $\mu(1) = 1$, $\mu(n) = (-1)^k$ if $n$ is a product of $k$ distinct primes, and $\mu(n) = 0$ if $n$ has a squared prime factor
  • The Euler totient function $\phi(n)$
    • Counts the number of positive integers up to $n$ that are relatively prime to $n$
  • The divisor function $d(n)$
    • Counts the number of positive divisors of $n$
  • The sum of divisors function $\sigma(n)$
    • Gives the sum of all positive divisors of $n$

Convolution Identities

• Many arithmetic functions are related through convolution identities
• The Möbius inversion formula is a key example:
  • $g(n) = \sum_{d|n} f(d) \Leftrightarrow f(n) = \sum_{d|n} \mu(d)g(n/d)$
  • This formula allows for the inversion of sums involving arithmetic functions

Applications of Arithmetic Functions

Simplifying Expressions and Deriving Identities

• Use the properties of additive and multiplicative functions to simplify expressions and derive identities
  • For example, if $f(n)$ and $g(n)$ are multiplicative, then $f(n)g(n)$ is also multiplicative
• Apply the Möbius inversion formula to invert sums involving arithmetic functions
  • This can be used to express one arithmetic function in terms of another

Solving Number-Theoretic Problems

• Arithmetic functions can be used to solve problems related to the distribution of prime numbers
  • For instance, proving the infinitude of primes using Euler's product formula for the Riemann zeta function: $\zeta(s) = \prod_{p \text{ prime}} (1 - p^{-s})^{-1}$
• Use arithmetic functions to derive asymptotic estimates for number-theoretic quantities
  • Such as the average order of the divisor function $d(n)$
• Employ arithmetic functions in the study of multiplicative number theory
  • For example, in the proof of the Erdős-Kac theorem on the normal distribution of the prime factors of integers

Analyzing Arithmetic Functions with Dirichlet Series

Definitions and Properties

• Dirichlet series are infinite series of the form $\sum_{n=1}^\infty \frac{a_n}{n^s}$, where $s$ is a complex variable and $a_n$ is a sequence of complex numbers
• Many arithmetic functions have associated Dirichlet series
  • For example, the Riemann zeta function $\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}$ for the constant function $a_n = 1$
• The Dirichlet series of a multiplicative function has an Euler product representation
  • This can be used to study its analytic properties
• The Dirichlet series of an additive function can be expressed in terms of the Dirichlet series of related multiplicative functions using convolution identities

Analytic Properties and Asymptotic Behavior

• Analytic properties of Dirichlet series provide insights into the asymptotic behavior of the associated arithmetic functions
  • These properties include their meromorphic continuation and location of poles and zeros
• The Wiener-Ikehara theorem relates the asymptotic behavior of an arithmetic function to the analytic properties of its Dirichlet series
  • This enables the derivation of precise estimates for the growth of arithmetic functions
• Studying the analytic properties of Dirichlet series can lead to important results in number theory
  • Such as the Prime Number Theorem, which describes the asymptotic distribution of prime numbers