💠Intro to Complex Analysis Unit 11 – The Riemann Zeta Function

Definition and Basics

• The Riemann zeta function is a complex-valued function denoted as $\zeta(s)$ where $s$ is a complex number
• Defined for complex numbers $s$ with real part greater than 1 by the infinite series $\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s} = \frac{1}{1^s} + \frac{1}{2^s} + \frac{1}{3^s} + \cdots$
• Converges absolutely for all complex numbers $s$ with real part greater than 1
• Can be extended to a meromorphic function defined on the whole complex plane, with a simple pole at $s=1$
• Satisfies the functional equation $\zeta(s) = 2^s \pi^{s-1} \sin\left(\frac{\pi s}{2}\right) \Gamma(1-s) \zeta(1-s)$, linking values of $\zeta(s)$ at $s$ and $1-s$
  • $\Gamma(s)$ denotes the gamma function, a generalization of the factorial function to complex numbers
• Has trivial zeros at the negative even integers $(-2, -4, -6, \ldots)$ due to the sine function in the functional equation
• Plays a crucial role in analytic number theory and has deep connections to the distribution of prime numbers

Historical Context

• Leonhard Euler first introduced the zeta function in the 18th century while studying infinite series
  • Euler proved the Basel problem, showing that $\zeta(2) = \frac{\pi^2}{6}$
• Bernhard Riemann's 1859 paper "On the Number of Primes Less Than a Given Magnitude" significantly expanded the theory of the zeta function
• Riemann extended the zeta function to the complex plane using analytic continuation
• Formulated the Riemann hypothesis, conjecturing that all non-trivial zeros of the zeta function have real part equal to $\frac{1}{2}$
• The Riemann hypothesis remains one of the most famous unsolved problems in mathematics
  • Its resolution would have profound implications for the distribution of prime numbers and many other areas of mathematics
• Throughout the 20th and 21st centuries, mathematicians have continued to study the zeta function, uncovering new properties and connections to various branches of mathematics

Key Properties

• The zeta function has a simple pole at $s=1$ with residue 1
• Satisfies the Euler product formula $\zeta(s) = \prod_{p \text{ prime}} \frac{1}{1-p^{-s}}$ for $\Re(s) > 1$, linking the zeta function to prime numbers
• Has a functional equation relating values at $s$ and $1-s$, which can be used to extend the zeta function to the entire complex plane
• Non-trivial zeros of the zeta function are symmetric about the critical line $\Re(s) = \frac{1}{2}$ due to the functional equation
• Satisfies various identities and relations, such as the reflection formula and the Riemann-Siegel formula
• Values at positive even integers are related to Bernoulli numbers: $\zeta(2n) = (-1)^{n+1} \frac{(2\pi)^{2n}}{2(2n)!} B_{2n}$ for $n \geq 1$
• Values at positive odd integers are not known to have a closed form, but can be expressed in terms of the Riemann-Siegel theta function
• Has a deep connection to the distribution of prime numbers through the prime number theorem and related results

Analytical Continuation

• The zeta function, initially defined as a series for $\Re(s) > 1$, can be extended to a meromorphic function on the entire complex plane
• Riemann used the contour integral representation $\zeta(s) = \frac{1}{2\pi i} \int_C \frac{(-x)^s}{(e^x - 1)x} \, dx$ to perform the analytic continuation
  • The contour $C$ encircles the positive real axis counterclockwise, avoiding the origin and the point at infinity
• The functional equation $\zeta(s) = 2^s \pi^{s-1} \sin\left(\frac{\pi s}{2}\right) \Gamma(1-s) \zeta(1-s)$ provides another way to extend the zeta function
• Analytic continuation allows the study of the zeta function's properties and behavior in regions where the original series definition is not valid
• Enables the investigation of the zeta function's zeros, particularly the non-trivial zeros in the critical strip $0 < \Re(s) < 1$
• Facilitates the derivation of various identities and relations involving the zeta function, such as the reflection formula and the Riemann-Siegel formula

Zeros and the Riemann Hypothesis

• The Riemann zeta function has two types of zeros: trivial zeros and non-trivial zeros
• Trivial zeros occur at the negative even integers $(-2, -4, -6, \ldots)$ due to the sine function in the functional equation
• Non-trivial zeros are complex numbers $s$ with $0 < \Re(s) < 1$ that satisfy $\zeta(s) = 0$
  • The functional equation implies that non-trivial zeros are symmetric about the critical line $\Re(s) = \frac{1}{2}$
• The Riemann hypothesis, proposed by Bernhard Riemann in 1859, states that all non-trivial zeros of the zeta function have real part equal to $\frac{1}{2}$
  • In other words, the hypothesis asserts that all non-trivial zeros lie on the critical line
• The Riemann hypothesis is one of the most important unsolved problems in mathematics, with far-reaching consequences in number theory and other areas
• Extensive computational evidence supports the Riemann hypothesis, with billions of non-trivial zeros calculated and found to lie on the critical line
• The distribution of the zeta function's zeros is closely related to the distribution of prime numbers and the behavior of various arithmetic functions

Applications in Number Theory

• The Riemann zeta function has numerous applications in analytic number theory, particularly in the study of prime numbers and arithmetic functions
• The prime number theorem, which describes the asymptotic distribution of prime numbers, can be proved using the zeta function
  • The theorem states that the number of primes less than or equal to $x$ is asymptotically equal to $\frac{x}{\log x}$ as $x \to \infty$
• The Riemann hypothesis, if true, would provide a more precise estimate for the error term in the prime number theorem
• The zeta function appears in the explicit formulas for various arithmetic functions, such as the Möbius function and the von Mangoldt function
• Generalizations of the zeta function, such as Dirichlet L-functions and Dedekind zeta functions, are used to study the distribution of prime ideals in algebraic number fields
• The Riemann zeta function is connected to the Riemann-Siegel formula, which provides an efficient way to calculate the zeta function on the critical line
• The behavior of the zeta function near $s=1$ is related to the Mertens conjecture and the Meissel-Mertens constant, which concern the growth of sums of the Möbius function

Computational Methods

• Computing values of the Riemann zeta function is crucial for numerical investigations and testing conjectures related to its properties
• For $\Re(s) > 1$, the zeta function can be computed using the defining series $\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}$, although convergence can be slow for $s$ close to 1
• The Euler-Maclaurin formula provides a more efficient method for computing the zeta function, expressing it as a sum of an integral and correction terms
• The Riemann-Siegel formula is particularly useful for calculating values of the zeta function on the critical line $\Re(s) = \frac{1}{2}$
  • It expresses $\zeta\left(\frac{1}{2} + it\right)$ as a sum of two rapidly converging series, enabling efficient computation
• Contour integration methods, such as the Odlyzko-Schönhage algorithm, can be used to calculate the zeta function to high precision
• Approximations and asymptotic expansions, such as the Stirling approximation for the gamma function, are often employed in computational methods
• Efficient algorithms for locating and verifying the zeta function's zeros are essential for numerical studies related to the Riemann hypothesis

Related Functions and Generalizations

• The Riemann zeta function is a special case of a broader class of functions called L-functions, which share similar properties and are central to modern analytic number theory
• Dirichlet L-functions, denoted as $L(s, \chi)$, are generalizations of the zeta function that incorporate Dirichlet characters $\chi$
  • They are used to study the distribution of primes in arithmetic progressions and the properties of Dirichlet characters
• Dedekind zeta functions, associated with algebraic number fields, are analogues of the Riemann zeta function for number fields
  • They encode information about the prime ideals and the arithmetic structure of the number field
• The Hurwitz zeta function $\zeta(s, a)$ is a generalization of the Riemann zeta function that includes an additional parameter $a$
  • It reduces to the Riemann zeta function when $a=1$ and has applications in analytic number theory and other areas of mathematics
• Polylogarithms, which generalize the logarithm function, are closely related to the zeta function and appear in various branches of mathematics and physics
• Multiple zeta values, also known as Euler-Zagier sums, are generalizations of the zeta function that involve sums of products of powers of integers
  • They have connections to knot theory, quantum field theory, and other areas of mathematics and physics