Glossary

7.2 Analytic continuation of the zeta function

3 min read•august 9, 2024

The is a cornerstone of analytic number theory. This section explores how to extend its definition beyond real numbers greater than 1. We'll see how techniques allow us to understand the zeta function's behavior across the entire complex plane.

is a powerful tool that reveals hidden properties of functions. For the zeta function, this process uncovers fascinating connections to prime numbers and opens doors to studying its zeros, including the famous .

Analytic Continuation and Complex Analysis

Extending Functions Beyond Their Original Domain

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  • Analytic continuation expands function definitions beyond initial domains
  • Complex analysis provides tools for studying functions in complex plane
  • possess complex differentiability in a neighborhood
  • representations enable function extension to larger domains
  • ensures consistent continuation along different paths
  • visualize multi-valued functions (square root function)

Meromorphic Functions and Laurent Series

  • behave analytically except at isolated
  • generalizes power series for functions with singularities
  • Series expansion consists of positive and negative power terms
  • contains negative power terms, describes behavior near singularity
  • contains non-negative power terms, represents regular behavior
  • Laurent series aids in classifying singularities (removable, , essential)

Poles and Residues

Understanding Poles in Complex Analysis

  • Poles represent isolated singularities where function values approach infinity
  • Order of pole determined by leading term in Laurent series expansion
  • Simple poles have order 1, characterized by 1z−a\frac{1}{z-a} term
  • Higher-order poles involve higher negative powers of (z−a)(z-a)
  • Poles crucial for understanding behavior of meromorphic functions
  • Visualize poles as "punctures" in the complex plane where function blows up

Residues and Their Applications

  • quantify behavior of functions near poles
  • Calculated as coefficient of 1z−a\frac{1}{z-a} term in Laurent expansion
  • connects contour integrals to sum of residues
  • Applications in evaluating real integrals (Fourier transforms, improper integrals)
  • Residues used in signal processing, control theory, and quantum mechanics

The Reflection Formula and Functional Equations

  • relates function values at symmetric points
  • For zeta function: ζ(1−s)=2(2π)−scos⁡(πs2)Γ(s)ζ(s)\zeta(1-s) = 2(2\pi)^{-s}\cos(\frac{\pi s}{2})\Gamma(s)\zeta(s)
  • describe symmetries and relationships in special functions
  • Provide insights into function behavior across entire complex plane
  • Used to extend definitions and prove properties of analytic continuations
  • Reflection principles apply to broader class of functions (Schwarz reflection)

Special Functions

The Gamma Function and Its Properties

  • extends factorial to complex numbers
  • Defined as Γ(z)=∫0∞tz−1e−tdt\Gamma(z) = \int_0^\infty t^{z-1}e^{-t}dt for Re(z) > 0
  • Satisfies functional equation Γ(z+1)=zΓ(z)\Gamma(z+1) = z\Gamma(z)
  • Analytic continuation extends definition to entire complex plane (except negative integers)
  • Poles at non-positive integers with residues (−1)nn!\frac{(-1)^n}{n!}
  • Relates to zeta function through functional equation
  • Applications in probability theory, statistical mechanics, and quantum physics
  • connects gamma and zeta functions
  • provides asymptotic behavior for large arguments

Key Terms to Review (25)

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.
Analytic part: The analytic part refers to the component of a mathematical function or object that is described in terms of convergence, continuity, and differentiability. It connects to important concepts such as analytic functions, which are functions that can be represented by a power series in a neighborhood of every point in their domain, highlighting their smoothness and well-behaved nature.
Complex Analysis: Complex analysis is a branch of mathematics that studies functions of complex numbers and their properties, focusing on the behavior of these functions in the complex plane. It plays a critical role in understanding various aspects of number theory, including the distribution of prime numbers and the behavior of special functions like the Riemann zeta function.
Contour Integration: Contour integration is a method in complex analysis used to evaluate integrals along a path or contour in the complex plane. This technique is particularly useful in analytic number theory, as it helps establish connections between functions like the Riemann zeta function and the distribution of prime numbers.
Essential Singularity: An essential singularity is a point at which a complex function behaves in a highly unpredictable manner, lacking any limit or well-defined value as it approaches that point. Unlike poles, where the function tends to infinity, or removable singularities, where the function can be redefined, essential singularities are characterized by their chaotic behavior, resulting in various possible values in every neighborhood of the singularity. This concept is crucial for understanding the analytic continuation of functions like the zeta function.
Functional equations: Functional equations are equations that specify a relationship between the values of a function at different points. They often arise in various mathematical contexts, including number theory, where they help uncover properties of number-theoretic functions like the Riemann zeta function. These equations can provide insights into the analytic properties and behavior of functions, which is essential for proving important theorems, such as the distribution of prime numbers.
Gamma Function: The Gamma function is a mathematical function that extends the concept of factorials to complex and real number arguments. It is defined for positive real numbers and is denoted as $$ ext{Γ}(n) = (n-1)!$$ for natural numbers. The Gamma function plays a crucial role in various areas of mathematics, including number theory, calculus, and complex analysis, often connecting to concepts like the Riemann zeta function and analytic continuation.
Holomorphic functions: Holomorphic functions are complex functions that are differentiable at every point in their domain. This differentiability implies that these functions can be represented by a power series, which converges within a certain radius. Holomorphic functions play a crucial role in complex analysis and have significant applications in number theory, particularly in the analytic continuation of functions and proofs of arithmetic theorems.
Laurent Series: A Laurent series is a representation of a complex function as a power series that includes both positive and negative powers of the variable. This type of series is particularly useful in complex analysis, especially when dealing with functions that have singularities, allowing for the expansion of functions in annular regions. The presence of negative powers distinguishes Laurent series from Taylor series and is essential in understanding the behavior of functions around their singular points.
Mellin Transform: The Mellin transform is a powerful integral transform that converts functions defined on the positive real line into functions defined on the complex plane, often used in number theory and analysis. It establishes a bridge between the algebra of functions and their multiplicative properties, making it especially useful in deriving properties of Dirichlet series and in understanding the distribution of prime numbers.
Meromorphic functions: Meromorphic functions are complex functions that are holomorphic (analytic) everywhere except at a discrete set of isolated points, known as poles, where they may approach infinity. These functions are significant in complex analysis as they generalize the concept of rational functions, allowing for a broader understanding of analytic properties. Their behavior around poles and the possibility of extending them to larger domains is crucial in various fields, especially when dealing with functions like the Riemann zeta function and in proving arithmetic theorems.
Poles: In the context of analytic number theory, poles refer to certain points in the complex plane where a meromorphic function, such as the Riemann zeta function, becomes undefined or diverges to infinity. The behavior of a function near its poles is crucial for understanding its properties, including analytic continuation and non-vanishing values on specific lines in the complex plane.
Power Series: A power series is an infinite series of the form $$ ext{a}_0 + ext{a}_1x + ext{a}_2x^2 + ext{a}_3x^3 + ...$$ where $$ ext{a}_n$$ represents the coefficients and $$x$$ is a variable. Power series are essential in analyzing functions, particularly in the context of analytic continuation, as they allow for the representation of complex functions within their radius of convergence. These series can provide insights into function behavior and facilitate calculations in various branches of mathematics.
Prime Number Theorem: The Prime Number Theorem describes the asymptotic distribution of prime numbers, stating that the number of primes less than a given number $n$ is approximately $\frac{n}{\log(n)}$. This theorem establishes a connection between primes and logarithmic functions, which has far-reaching implications in analytic number theory, especially in understanding the distribution of primes and their density among integers.
Principal Part: The principal part refers to the most significant or dominant component of a function or expression, particularly in complex analysis and analytic continuation. In the context of the zeta function, the principal part helps in understanding the behavior of the function near its poles and singularities, allowing for the extension of the function's domain beyond its original limits.
Reflection formula: The reflection formula is an important identity that connects the values of the zeta function at $s$ and $1-s$. This relationship not only highlights the symmetry of the zeta function but also serves as a crucial tool in understanding its analytic continuation beyond its initial domain. The formula is instrumental in revealing properties of the zeta function and plays a key role in number theory, especially in the study of prime numbers and their distribution.
Removable singularity: A removable singularity is a type of isolated singularity of a complex function where the function can be defined at that point such that it becomes analytic. This means that if the singularity is 'removed' by redefining the function at that point, the new function is still analytic in some neighborhood around it. This concept is crucial in understanding the analytic continuation of functions, as it allows for extending the domain of those functions.
Residue Theorem: The Residue Theorem is a powerful tool in complex analysis that allows for the evaluation of complex line integrals by relating them to the residues of singular points within a closed contour. This theorem connects to various important concepts in number theory, particularly in understanding the behavior of Dirichlet series and analytic functions.
Residues: In complex analysis, residues are complex numbers that represent the coefficients of the terms in the Laurent series expansion of a function around a singularity. They play a critical role in evaluating integrals and understanding the behavior of functions at poles, particularly in the context of the analytic continuation of the zeta function.
Riemann Hypothesis: The Riemann Hypothesis is a conjecture in number theory that states all non-trivial zeros of the Riemann zeta function lie on the critical line in the complex plane, where the real part of s is 1/2. This hypothesis is crucial as it connects the distribution of prime numbers to the properties of analytic functions, influencing various aspects of number theory and its applications.
Riemann surfaces: Riemann surfaces are one-dimensional complex manifolds that allow for the multi-valued functions of complex analysis to be treated as single-valued objects. They provide a natural setting for understanding complex functions and their singularities, particularly in the context of analytic continuation of functions like the zeta function, facilitating the study of their properties across different domains.
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.
Singular Points: Singular points refer to locations in a complex function where the function ceases to be analytic, often manifesting as poles or essential singularities. Understanding singular points is crucial for studying the analytic continuation of functions like the zeta function, as they dictate the behavior and properties of the function in regions where it cannot be expressed by a power series.
Stirling's Approximation: Stirling's approximation is a formula used to estimate the factorial of a large number, providing an asymptotic approximation that simplifies calculations involving factorials. It expresses the factorial in terms of simpler functions, making it particularly useful in various areas of mathematics, including the analysis of series and the study of analytic properties of functions like the zeta function. The approximation reveals connections to logarithmic and exponential functions, especially when analyzing growth rates and convergence behaviors.
Uniqueness Theorem: The uniqueness theorem in the context of analytic continuation states that if a function is meromorphic on a domain and it is equal to another meromorphic function on a set with a limit point within that domain, then the two functions must be identical everywhere in that domain. This theorem is crucial for understanding how the Riemann zeta function can be analytically continued beyond its initial definition, establishing the fundamental connection between the values of this function at different points.
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