6.1 Introduction to Dirichlet series

Dirichlet series are powerful tools in number theory, representing functions as infinite sums. They're key to studying prime numbers and other arithmetic properties. This intro to Dirichlet series lays the groundwork for understanding their form and convergence.

We'll explore the general form of Dirichlet series, their convergence properties, and the important concept of the abscissa of convergence. These ideas are crucial for grasping how Dirichlet series behave and where they're useful in number theory.

Definition and Convergence of Dirichlet Series

Understanding Dirichlet Series and Their General Form

• Dirichlet series represent a class of infinite series in complex analysis and number theory
• General form of a Dirichlet series expressed as $\sum_{n=1}^{\infty} \frac{a_n}{n^s}$
• $a_n$ denotes a sequence of complex numbers
• $s$ represents a complex variable
• Dirichlet series generalize power series by replacing $x^n$ with $n^{-s}$
• Applications include studying arithmetic functions and analytic properties of L-functions
• Named after Peter Gustav Lejeune Dirichlet, a 19th-century German mathematician

Convergence Concepts for Dirichlet Series

• Abscissa of convergence defines the boundary of convergence for a Dirichlet series
• Represents the infimum of real parts of $s$ for which the series converges
• Denoted as $\sigma_c$, where $\sigma = \text{Re}(s)$
• Series converges for all $s$ with $\text{Re}(s) > \sigma_c$
• Abscissa of absolute convergence determines where the series converges absolutely
• Denoted as $\sigma_a$, where $\sigma_a \geq \sigma_c$
• Series converges absolutely for all $s$ with $\text{Re}(s) > \sigma_a$

Half-Plane of Convergence and Its Significance

• Half-plane of convergence describes the region where a Dirichlet series converges
• Defined as the set of complex numbers $s$ where $\text{Re}(s) > \sigma_c$
• Visualized as the right half of the complex plane, bounded by a vertical line
• Convergence behavior changes across this boundary
• Important for understanding the analytic properties of the series
• Helps determine the domain of definition for functions represented by Dirichlet series
• Crucial in the study of analytic continuation and functional equations

Analytic Properties and Special Functions

Analytic Continuation of Dirichlet Series

• Analytic continuation extends the domain of a function beyond its original region of definition
• Allows Dirichlet series to be studied in regions where they may not initially converge
• Preserves the analytic properties of the original function
• Enables the exploration of zeros and poles of the extended function
• Reveals important connections between different mathematical objects
• Technique involves finding an analytic function that agrees with the original series in its domain of convergence
• Crucial for understanding the behavior of Dirichlet series in the complex plane

The Riemann Zeta Function and Its Properties

• Riemann zeta function represents the most famous example of a Dirichlet series
• Defined as $\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}$ for $\text{Re}(s) > 1$
• Converges absolutely for $\text{Re}(s) > 1$ and conditionally for $\text{Re}(s) > 0$
• Analytically continued to the entire complex plane except for a simple pole at $s = 1$
• Satisfies the functional equation $\zeta(s) = 2^s \pi^{s-1} \sin(\frac{\pi s}{2}) \Gamma(1-s) \zeta(1-s)$
• Zeros of the zeta function play a crucial role in the distribution of prime numbers
• Connected to the Riemann Hypothesis, one of the most important unsolved problems in mathematics

Dirichlet L-Functions and Their Significance

• Dirichlet L-functions generalize the Riemann zeta function
• Associated with Dirichlet characters, which are certain arithmetic functions
• General form given by $L(s, \chi) = \sum_{n=1}^{\infty} \frac{\chi(n)}{n^s}$
• $\chi$ represents a Dirichlet character
• Play a crucial role in studying the distribution of prime numbers in arithmetic progressions
• Used in proving Dirichlet's theorem on primes in arithmetic progressions
• Exhibit analytic continuation and functional equations similar to the Riemann zeta function
• Important in the study of class field theory and algebraic number theory
• Generalized L-functions encompass a wide range of mathematical objects (elliptic curves, modular forms)