10.2 Dirichlet's theorem on primes in arithmetic progressions
3 min read•august 9, 2024
on primes in arithmetic progressions is a cornerstone of analytic number theory. It proves there are infinitely many primes in sequences like 3, 7, 11, 15, ... where each term differs by a fixed amount.
The proof hinges on and , which generalize ideas from the . By showing L(1,χ) ≠ 0 for , Dirichlet established his groundbreaking result on prime distributions.
Dirichlet Characters and L-functions
Fundamental Concepts of Dirichlet Characters
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- Dirichlet characters define completely multiplicative functions from integers to complex numbers
- Characters modulo q map integers with the same residue modulo q to the same value
- χ₀ equals 1 for integers to q and 0 otherwise
- Non-principal characters have the property that the sum of their values over a complete set of residues modulo q equals 0
- for Dirichlet characters play a crucial role in many proofs
L-functions and Their Properties
- L-functions associated with Dirichlet characters generalize the Riemann zeta function
- For a Dirichlet character χ, the L-function L(s, χ) defined as the sum of χ(n)/n^s for positive integers n
- L-functions converge absolutely for Re(s) > 1
- expresses L(s, χ) as a product over primes: L(s, χ) = ∏ᵨ (1 - χ(p)/p^s)^(-1)
- extends L(s, χ) to the entire complex plane for non-principal characters
- For the principal character, L(s, χ₀) has a simple pole at s = 1, similar to the Riemann zeta function
Dirichlet's Theorem and Non-vanishing of L(1, χ)
Dirichlet's Theorem on Primes in Arithmetic Progressions
- States that for coprime positive integers a and q, there exist infinitely many primes in the a + nq
- Proof relies on showing that L(1, χ) ≠ 0 for all non-principal characters modulo q
- Utilizes the of L(s, χ) to relate primes in arithmetic progressions to L-functions
- Demonstrates that the in each reduced residue class modulo q approaches 1/φ(q) as x approaches infinity
Non-vanishing of L(1, χ) and Its Implications
- L(1, χ) ≠ 0 for all non-principal characters χ modulo q forms a crucial step in proving Dirichlet's theorem
- Proof involves showing that the product of L(1, χ) over all characters χ modulo q is non-zero
- Utilizes properties of and the for L-functions
- Non-vanishing of L(1, χ) has implications beyond Dirichlet's theorem, including in the study of class numbers of number fields
Functional Equation for L-functions
- Relates values of L(s, χ) in the left half-plane to those in the right half-plane
- For χ modulo q, the functional equation involves Gauss sums and gamma functions
- Takes the form L(1-s, χ̄) = W(χ)(q/π)^(s-1/2) Γ(s/2) L(s, χ) / Γ((1-s)/2), where W(χ) denotes a certain complex number of absolute value 1
- Allows for the study of L-functions in regions where the original series definition does not converge
- Plays a crucial role in the analytic continuation of L-functions and in proving the non-vanishing of L(1, χ)
Generalized Riemann Hypothesis
Statement and Implications of the Generalized Riemann Hypothesis
- Conjectures that all non-trivial zeros of L(s, χ) lie on the Re(s) = 1/2 for all Dirichlet characters χ
- Generalizes the classical Riemann Hypothesis from the Riemann zeta function to all Dirichlet L-functions
- Would imply strong results about the distribution of primes in arithmetic progressions
- Provides upper bounds for the least prime in an arithmetic progression a + nq ()
- Impacts various areas of number theory, including estimates for and the distribution of
Connections to Other Mathematical Problems
- Relates to the Extended Riemann Hypothesis for zeta functions of number fields
- Influences the study of elliptic curves and their L-functions
- Connects to problems in computational number theory, such as primality testing and factorization algorithms
- Impacts the understanding of the distribution of Gaussian primes in the complex plane
- Provides insights into the behavior of multiplicative functions in number theory
Key Terms to Review (22)
Analytic Continuation: Analytic continuation is a technique in complex analysis that extends the domain of a given analytic function beyond its original area of definition, allowing it to be expressed in a broader context. This process is crucial for understanding functions like the Riemann zeta function and Dirichlet L-functions, as it reveals their behavior and properties in different regions of the complex plane.
Arithmetic Progression: An arithmetic progression is a sequence of numbers in which the difference between consecutive terms is constant. This common difference can be positive, negative, or zero, and it helps form a linear pattern that is essential in various areas of number theory, particularly when studying primes. Understanding arithmetic progressions is crucial when investigating the distribution of prime numbers within these sequences and analyzing their properties through advanced theorems.
Character sums: Character sums are mathematical expressions that sum values assigned to elements of a group, often related to number theory and modular arithmetic. These sums play a crucial role in studying the distribution of prime numbers, particularly when analyzing primes in specific arithmetic progressions and in evaluating Dirichlet series. Understanding character sums helps connect various aspects of analytic number theory, especially in revealing properties of integers through their residues modulo some integer.
Coprime: Coprime, or relatively prime, refers to two integers that have no common positive integer factors other than 1. This concept is essential in number theory, particularly when discussing the distribution of prime numbers and properties related to modular arithmetic, as it helps determine the conditions under which certain sequences yield prime numbers.
Critical Line: The critical line refers to the vertical line in the complex plane defined by the equation Re(s) = 1/2, where s is a complex number. This line is significant in the study of the Riemann zeta function and its properties, particularly concerning the distribution of prime numbers and the famous Riemann Hypothesis.
Density of primes: The density of primes refers to the concept of how the prime numbers are distributed among the integers, often evaluated in terms of their asymptotic behavior as we consider larger and larger numbers. This idea is key in understanding various number-theoretic functions, which help analyze how frequently primes appear in specified sets or sequences, particularly when discussing properties such as arithmetic progressions or applying sieve methods.
Dirichlet Characters: Dirichlet characters are certain completely multiplicative arithmetic functions that are defined modulo some integer $k$. They play a crucial role in number theory, particularly in the study of primes in arithmetic progressions and the properties of Dirichlet L-functions. These characters help to generalize the concept of classical characters in modular arithmetic and provide essential insights into the distribution of primes.
Dirichlet Density: Dirichlet density is a way to measure the 'size' of a set of integers that share a particular property, often concerning primes in arithmetic progressions. It captures the idea of how many integers in that set exist relative to the total integers in a specified range, offering insights into the distribution of those integers. This concept is especially relevant when analyzing the distribution of prime numbers in sequences defined by linear functions, which connects deeply to results about primes in specific arithmetic progressions.
Dirichlet's Theorem: Dirichlet's Theorem states that there are infinitely many prime numbers in any arithmetic progression of the form $$a + nd$$, where $$a$$ and $$d$$ are coprime integers, and $$n$$ is a non-negative integer. This theorem connects the distribution of primes to arithmetic progressions, revealing that primes are not just isolated occurrences but instead occur regularly in structured patterns.
Euler product representation: The Euler product representation is a way to express a Dirichlet series as an infinite product over prime numbers, linking the distribution of prime numbers to the properties of arithmetic functions. This representation reveals how the primes influence various number-theoretic functions and plays a crucial role in understanding the distribution of primes within arithmetic progressions, highlighting the deep connection between primes and analytic properties.
Functional Equation: A functional equation is a mathematical equation that specifies a relationship between the values of a function at different points. These equations often arise in number theory and analysis, linking various properties of functions like the Riemann zeta function or Dirichlet series, helping us understand their behavior across different domains.
Gauss Sums: Gauss sums are special types of exponential sums that play a significant role in number theory, particularly in relation to quadratic residues and character sums. These sums are often expressed in the form $$G(a, p) = \sum_{n=0}^{p-1} e^{2\pi i \frac{an^2}{p}}$$ for integers $a$ and prime $p$. Gauss sums are essential for understanding the distribution of prime numbers in arithmetic progressions and are deeply connected to Dirichlet's theorem, which asserts that there are infinitely many primes in specific arithmetic sequences.
Generalized Riemann Hypothesis: The Generalized Riemann Hypothesis (GRH) extends the classical Riemann Hypothesis to Dirichlet L-functions, asserting that all non-trivial zeros of these functions lie on the critical line in the complex plane, which is given by the real part being equal to 1/2. This hypothesis plays a significant role in number theory, particularly in understanding the distribution of prime numbers in arithmetic progressions and various properties of Dirichlet characters and L-functions.
L-functions: L-functions are complex functions that arise in number theory and are closely related to the distribution of prime numbers and other number-theoretic objects. They generalize the Riemann zeta function and are central in various results, including those about primes in arithmetic progressions, Dirichlet characters, and the implications of the Riemann Hypothesis.
Linnik's Constant: Linnik's constant is a number, denoted as L, that appears in the context of prime numbers, specifically relating to Dirichlet's theorem on primes in arithmetic progressions. This constant represents the upper bound on the size of the prime gaps in arithmetic progressions, specifically indicating that there are infinitely many primes within these sequences under certain conditions. The existence of Linnik's constant provides insight into the distribution of prime numbers and enhances our understanding of their behavior in modular forms.
Logarithmic Derivative: The logarithmic derivative of a function is defined as the derivative of the logarithm of that function, providing useful information about growth rates and multiplicative structures. This concept is particularly relevant in number theory, as it helps analyze functions like the Riemann zeta function and Euler's totient function, especially in contexts involving primes and arithmetic progressions.
Non-Principal Characters: Non-principal characters are specific types of Dirichlet characters that are not induced by the principal character modulo a given integer. They play a crucial role in understanding the distribution of primes in arithmetic progressions, as they help determine how prime numbers are distributed across different residue classes. Non-principal characters have unique properties that affect the behavior of L-functions and are essential in the application of Dirichlet's theorem.
Orthogonality Relations: Orthogonality relations refer to the mathematical conditions that describe the independence of different Dirichlet characters and their associated L-functions. These relations indicate that the inner product of two distinct characters over a certain set of integers is zero, highlighting their distinct contributions. This concept is essential for understanding the distribution of primes in arithmetic progressions and how Dirichlet characters can be utilized to form nontrivial results in analytic number theory.
Primitive characters: Primitive characters are specific types of Dirichlet characters that are defined modulo a positive integer $q$ and cannot be expressed as the product of a character of lower order. They play a crucial role in number theory, particularly in understanding the distribution of prime numbers in arithmetic progressions. These characters help establish results like Dirichlet's theorem, which asserts that there are infinitely many primes in arithmetic progressions, as long as the conditions of coprimality between the common difference and the modulus are satisfied.
Principal Character: A principal character is a specific type of Dirichlet character that is associated with the trivial character modulo some integer, typically denoted as 1. This character takes the value of 1 for all integers coprime to the modulus and 0 for integers that are not coprime, serving as a fundamental building block in the study of Dirichlet characters and their properties. The principal character plays a crucial role in many aspects of analytic number theory, particularly in understanding the distribution of prime numbers within arithmetic progressions.
Quadratic Residues: Quadratic residues are integers that can be expressed as the square of an integer modulo a prime number. More specifically, if p is a prime, an integer a is considered a quadratic residue modulo p if there exists some integer x such that $$x^2 \equiv a \mod p$$. Understanding quadratic residues is crucial for exploring properties of prime numbers and their distribution, particularly when looking at how they appear in arithmetic progressions.
Riemann zeta function: The Riemann zeta function is a complex function defined for complex numbers, which plays a pivotal role in number theory, particularly in understanding the distribution of prime numbers. It is intimately connected to various aspects of analytic number theory, including the functional equation, Dirichlet series, and the famous Riemann Hypothesis that conjectures all non-trivial zeros of the function lie on the critical line in the complex plane.