🔢Analytic Number Theory Unit 6 – Dirichlet Series & Euler Products

Definition and Basic Properties

• Dirichlet series defined as a complex function $f(s) = \sum_{n=1}^{\infty} \frac{a_n}{n^s}$ where $s$ is a complex variable and $a_n$ is a sequence of complex numbers
• Converges for $s$ with real part greater than some real number $\sigma_c$ called the abscissa of convergence
• Dirichlet series can be added, subtracted, and multiplied term by term within their half-plane of convergence
  • Addition: $\sum_{n=1}^{\infty} \frac{a_n}{n^s} + \sum_{n=1}^{\infty} \frac{b_n}{n^s} = \sum_{n=1}^{\infty} \frac{a_n + b_n}{n^s}$
  • Multiplication: $\left(\sum_{n=1}^{\infty} \frac{a_n}{n^s}\right) \cdot \left(\sum_{n=1}^{\infty} \frac{b_n}{n^s}\right) = \sum_{n=1}^{\infty} \frac{c_n}{n^s}$ where $c_n = \sum_{d|n} a_d b_{n/d}$
• Dirichlet series can be differentiated and integrated term by term within their half-plane of convergence
• Special case when $a_n = 1$ for all $n$ yields the Riemann zeta function $\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}$
• Dirichlet characters $\chi(n)$ used to define Dirichlet L-functions $L(s, \chi) = \sum_{n=1}^{\infty} \frac{\chi(n)}{n^s}$

Convergence and Analytic Continuation

• Dirichlet series converges absolutely for $\Re(s) > \sigma_a$ (abscissa of absolute convergence) and conditionally for $\sigma_c \leq \Re(s) \leq \sigma_a$
• Abscissa of convergence $\sigma_c$ determined by the growth rate of the coefficients $a_n$
  • If $a_n = O(n^k)$ for some $k$, then $\sigma_c \leq k+1$
  • If $a_n = o(n^k)$ for all $k$, then $\sigma_c = -\infty$
• Analytic continuation extends the domain of a Dirichlet series beyond its half-plane of convergence
• Dirichlet series can be analytically continued to a meromorphic function on the entire complex plane
• Analytic continuation unique if it exists and preserves the functional equation and other properties of the original Dirichlet series
• Riemann zeta function $\zeta(s)$ and Dirichlet L-functions $L(s, \chi)$ can be analytically continued to the entire complex plane, with poles at $s=1$

Functional Equations

• Functional equations relate values of a Dirichlet series at different points in the complex plane
• Riemann zeta function satisfies the functional equation $\zeta(s) = 2^s \pi^{s-1} \sin\left(\frac{\pi s}{2}\right) \Gamma(1-s) \zeta(1-s)$
  • Relates values of $\zeta(s)$ at $s$ and $1-s$
  • Gamma function $\Gamma(s)$ is a generalization of the factorial function to complex numbers
• Dirichlet L-functions satisfy a similar functional equation involving Dirichlet characters and Gauss sums
• Functional equations used to study the behavior of Dirichlet series in the critical strip $0 < \Re(s) < 1$
• Provide a way to calculate values of a Dirichlet series in terms of its values at other points
• Play a crucial role in the proof of the prime number theorem and other important results in analytic number theory

Connection to Zeta Functions

• Riemann zeta function $\zeta(s)$ is the prototypical example of a Dirichlet series
• Dirichlet L-functions $L(s, \chi)$ generalize the Riemann zeta function by incorporating Dirichlet characters $\chi(n)$
  • Recover $\zeta(s)$ when $\chi(n) = 1$ for all $n$ (principal character)
• Dedekind zeta functions $\zeta_K(s)$ associated to number fields $K$ are Dirichlet series with coefficients related to the ideals of $K$
• Zeta functions encode information about the distribution of prime numbers and the structure of number fields
• Zeros of zeta functions (especially the Riemann zeta function) are deeply connected to the distribution of prime numbers
  • Riemann hypothesis states that all non-trivial zeros of $\zeta(s)$ have real part equal to $\frac{1}{2}$
• Generalizations of zeta functions (L-functions) associated to various mathematical objects (elliptic curves, modular forms, etc.) are central objects of study in modern number theory

Euler Products and Prime Factorization

• Euler product formula expresses a Dirichlet series as an infinite product over prime numbers
• For the Riemann zeta function, the Euler product formula is $\zeta(s) = \prod_{p \text{ prime}} \left(1 - \frac{1}{p^s}\right)^{-1}$
  • Convergent for $\Re(s) > 1$
  • Connects the zeta function to the distribution of prime numbers
• Dirichlet L-functions and other zeta functions have similar Euler product representations
• Euler product formulas reflect the unique factorization of integers into prime numbers
  • Fundamental theorem of arithmetic states that every positive integer can be uniquely expressed as a product of prime powers
• Studying the Euler product of a Dirichlet series provides insight into the multiplicative structure of the coefficients $a_n$
• Euler products used to prove the non-vanishing of zeta functions and L-functions at certain points (e.g., $\zeta(s) \neq 0$ for $\Re(s) > 1$)

Applications in Number Theory

• Dirichlet series and zeta functions are powerful tools for studying the distribution of prime numbers
• Prime number theorem states that the number of primes less than or equal to $x$ is asymptotic to $\frac{x}{\log x}$ as $x \to \infty$
  • Equivalent to the statement that $\zeta(s)$ has no zeros on the line $\Re(s) = 1$
• Dirichlet's theorem on arithmetic progressions states that for any coprime integers $a$ and $d$, there are infinitely many primes of the form $a + nd$
  • Proved using Dirichlet L-functions and their non-vanishing at $s=1$
• Generalized Riemann hypothesis (GRH) states that all non-trivial zeros of Dirichlet L-functions have real part equal to $\frac{1}{2}$
  • Implies strong bounds on the distribution of prime numbers in arithmetic progressions
• Dirichlet series used to study the average behavior of arithmetic functions (e.g., the divisor function $d(n)$, the sum of divisors function $\sigma(n)$, etc.)
• Connections to the theory of modular forms and elliptic curves through the study of L-functions

Key Theorems and Proofs

• Proof of the Euler product formula for the Riemann zeta function
  • Uses the fundamental theorem of arithmetic and the convergence of the Dirichlet series for $\Re(s) > 1$
• Proof of the functional equation for the Riemann zeta function
  • Involves the Poisson summation formula and the properties of the Gamma function
• Proof of Dirichlet's theorem on arithmetic progressions
  • Uses the non-vanishing of Dirichlet L-functions at $s=1$ and the orthogonality relations for Dirichlet characters
• Proof of the prime number theorem using the Riemann zeta function
  • Relies on the non-vanishing of $\zeta(s)$ on the line $\Re(s) = 1$ and the behavior of $\zeta(s)$ near $s=1$
• Proofs of the analytic continuation and functional equations for Dirichlet L-functions and other zeta functions
  • Involve techniques from complex analysis and the theory of modular forms
• Partial results towards the Riemann hypothesis and the generalized Riemann hypothesis
  • Deligne's proof of the Riemann hypothesis for zeta functions of algebraic varieties over finite fields
  • Zero-free regions for the Riemann zeta function and Dirichlet L-functions

Computational Techniques and Examples

• Techniques for numerically computing values of the Riemann zeta function and Dirichlet L-functions
  • Euler-Maclaurin summation formula
  • Riemann-Siegel formula for calculating $\zeta(\frac{1}{2} + it)$ for large $t$
• Algorithms for locating zeros of the Riemann zeta function and Dirichlet L-functions
  • Riemann-Siegel formula and the Odlyzko-Schönhage algorithm
  • Verified the Riemann hypothesis for the first $10^{13}$ zeros of $\zeta(s)$
• Examples of Dirichlet series and their properties
  • Riemann zeta function $\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}$
  • Dirichlet beta function $\beta(s) = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)^s}$, satisfies $\beta(s) = \left(1 - 2^{1-s}\right) \zeta(s)$
  • Dirichlet eta function $\eta(s) = \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^s}$, satisfies $\eta(s) = \left(1 - 2^{1-s}\right) \zeta(s)$
  • Dirichlet L-functions $L(s, \chi) = \sum_{n=1}^{\infty} \frac{\chi(n)}{n^s}$ for various Dirichlet characters $\chi$
• Computational verification of the functional equations and Euler product formulas for specific Dirichlet series
• Numerical evidence for the Riemann hypothesis and the generalized Riemann hypothesis