analytic number theory unit 2 study guides

Key Concepts and Definitions

• Arithmetic function $f(n)$ maps positive integers to complex numbers
• Multiplicative function $f(mn) = f(m)f(n)$ for coprime $m$ and $n$
• Completely multiplicative function $f(mn) = f(m)f(n)$ for all $m$ and $n$
• Additive function $f(mn) = f(m) + f(n)$ for coprime $m$ and $n$
• Completely additive function $f(mn) = f(m) + f(n)$ for all $m$ and $n$
• Dirichlet convolution $(f * g)(n) = \sum_{d|n} f(d)g(n/d)$ combines two arithmetic functions
• Dirichlet series $\sum_{n=1}^{\infty} \frac{f(n)}{n^s}$ generates an arithmetic function $f(n)$
• Möbius function $\mu(n)$ is $(-1)^k$ if $n$ is a product of $k$ distinct primes, and $0$ otherwise

Arithmetic Functions: Types and Properties

• Arithmetic functions include multiplicative, additive, and other special functions
  • Examples: Euler's totient function $\phi(n)$, divisor function $\sigma(n)$, Möbius function $\mu(n)$
• Multiplicative functions satisfy $f(1) = 1$ and $f(mn) = f(m)f(n)$ for coprime $m$ and $n$
• Completely multiplicative functions satisfy $f(1) = 1$ and $f(mn) = f(m)f(n)$ for all $m$ and $n$
  • Example: Euler's totient function $\phi(n)$ is multiplicative but not completely multiplicative
• Additive functions satisfy $f(mn) = f(m) + f(n)$ for coprime $m$ and $n$
• Completely additive functions satisfy $f(mn) = f(m) + f(n)$ for all $m$ and $n$
  • Example: $\log(n)$ is a completely additive function
• Many arithmetic functions can be expressed as Dirichlet series or generating functions

Multiplicative and Additive Functions

• Multiplicative functions are determined by their values on prime powers
  • For a prime $p$ and positive integer $k$, $f(p^k) = f(p)^k$ if $f$ is completely multiplicative
• Euler's product formula connects Dirichlet series of multiplicative functions to prime numbers
  • $\sum_{n=1}^{\infty} \frac{f(n)}{n^s} = \prod_p \left(1 + \frac{f(p)}{p^s} + \frac{f(p^2)}{p^{2s}} + \cdots\right)$ for $\text{Re}(s) > 1$
• Additive functions satisfy $f(mn) = f(m) + f(n)$ for coprime $m$ and $n$
• Completely additive functions are determined by their values on prime powers
  • For a prime $p$ and positive integer $k$, $f(p^k) = kf(p)$ if $f$ is completely additive
• Examples of multiplicative functions: $\phi(n)$, $\sigma(n)$, $\mu(n)$
• Examples of additive functions: $\omega(n)$ (number of distinct prime factors), $\Omega(n)$ (total number of prime factors)

Dirichlet Series and Generating Functions

• Dirichlet series $\sum_{n=1}^{\infty} \frac{f(n)}{n^s}$ generates an arithmetic function $f(n)$
  • Converges for $\text{Re}(s) > \sigma_c$, where $\sigma_c$ is the abscissa of convergence
• Generating functions $\sum_{n=0}^{\infty} f(n)x^n$ encode arithmetic functions as power series
• Dirichlet series and generating functions facilitate the study of arithmetic functions
  • Enable derivation of identities, asymptotic behavior, and average values
• Euler's product formula connects Dirichlet series of multiplicative functions to primes
• Examples: Riemann zeta function $\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}$, Dirichlet L-functions $L(s, \chi) = \sum_{n=1}^{\infty} \frac{\chi(n)}{n^s}$

Dirichlet Convolution: Definition and Properties

• Dirichlet convolution $(f * g)(n) = \sum_{d|n} f(d)g(n/d)$ combines two arithmetic functions $f$ and $g$
• Dirichlet convolution is commutative: $f * g = g * f$
• Dirichlet convolution is associative: $(f * g) * h = f * (g * h)$
• The constant function $\mathbf{1}(n) = 1$ for all $n$ is the identity under Dirichlet convolution
  • $f * \mathbf{1} = \mathbf{1} * f = f$ for any arithmetic function $f$
• Möbius inversion formula: If $f = g * \mathbf{1}$, then $g = f * \mu$, where $\mu$ is the Möbius function
• Dirichlet convolution of multiplicative functions is multiplicative
• Dirichlet series of a Dirichlet convolution is the product of the Dirichlet series of the individual functions
  • If $F(s) = \sum_{n=1}^{\infty} \frac{f(n)}{n^s}$ and $G(s) = \sum_{n=1}^{\infty} \frac{g(n)}{n^s}$, then $F(s)G(s) = \sum_{n=1}^{\infty} \frac{(f * g)(n)}{n^s}$

Applications of Dirichlet Convolution

• Dirichlet convolution is used to study the distribution of arithmetic functions
• Möbius inversion formula allows the inversion of sums over divisors
  • Example: $\sum_{d|n} \mu(d) = [n = 1]$, where $[P]$ is the Iverson bracket (1 if $P$ is true, 0 otherwise)
• Dirichlet convolution helps derive identities involving arithmetic functions
  • Example: $\phi * \mathbf{1} = \text{id}$, where $\text{id}(n) = n$ for all $n$
• Dirichlet series of a Dirichlet convolution facilitates the study of asymptotic behavior
• Applications in analytic number theory, such as the proof of the prime number theorem
• Dirichlet convolution is used in the study of multiplicative functions and their averages

Important Theorems and Proofs

• Möbius inversion formula: If $f = g * \mathbf{1}$, then $g = f * \mu$
  • Proof uses the fact that $\sum_{d|n} \mu(d) = [n = 1]$
• Euler's product formula for Dirichlet series of multiplicative functions
  • Proof relies on the fundamental theorem of arithmetic and properties of multiplicative functions
• Dirichlet's theorem on arithmetic progressions: For coprime $a$ and $d$, the arithmetic progression $a, a+d, a+2d, \ldots$ contains infinitely many primes
  • Proof uses Dirichlet L-functions and the non-vanishing of $L(1, \chi)$ for non-principal characters $\chi$
• Wirsing's theorem: The mean value of a positive multiplicative function $f$ satisfying $f(p^k) \leq c^k$ for some $c > 0$ is asymptotically equal to $\prod_p \left(1 + \frac{f(p)}{p} + \frac{f(p^2)}{p^2} + \cdots\right)$
  • Proof relies on the properties of Dirichlet series and Euler's product formula

Problem-Solving Techniques and Examples

• Identify the type of arithmetic function (multiplicative, additive, or other special function)
• Use properties of multiplicative and additive functions to simplify calculations
  • Example: If $f$ is multiplicative, $f(p^k) = f(p)^k$ for prime $p$ and positive integer $k$
• Apply Dirichlet convolution to derive identities or study the distribution of functions
  • Example: Show that $\sum_{d|n} \phi(d) = n$ using the identity $\phi * \mathbf{1} = \text{id}$
• Utilize Dirichlet series and generating functions to study asymptotic behavior
  • Example: Prove that $\sum_{n \leq x} \frac{\phi(n)}{n} = \frac{6}{\pi^2}x + O(\log x)$ using the Dirichlet series of $\phi(n)$
• Use Möbius inversion to invert sums over divisors
  • Example: If $f(n) = \sum_{d|n} g(d)$, find an expression for $g(n)$ in terms of $f(n)$
• Apply theorems like Dirichlet's theorem on arithmetic progressions or Wirsing's theorem to solve problems
  • Example: Prove that there are infinitely many primes of the form $4k+3$ using Dirichlet's theorem