14.1 Riemann zeta function and generalizations

The Riemann zeta function is a cornerstone of number theory, connecting complex analysis and prime numbers. It's defined as an infinite series that converges for certain complex numbers, but its true power comes from extending it to the whole complex plane.

This function's zeros hold the key to understanding prime number distribution. The Riemann Hypothesis, which predicts where these zeros lie, remains one of math's biggest unsolved puzzles, driving research in various areas of mathematics.

The Riemann Zeta Function

Definition and Convergence

Top images from around the web for Definition and Convergence

• 

  Euler product approximation of the Riemann zeta function - MathOverflow View original

  Is this image relevant?

• 

  Riemann zeta function - Wikipedia View original

  Is this image relevant?

• 

  File:Riemann zeta function absolute value.png - Wikimedia Commons View original

  Is this image relevant?

• 

  Euler product approximation of the Riemann zeta function - MathOverflow View original

  Is this image relevant?

• 

  Riemann zeta function - Wikipedia View original

  Is this image relevant?

1 of 3

Top images from around the web for Definition and Convergence

• 

  Euler product approximation of the Riemann zeta function - MathOverflow View original

  Is this image relevant?

• 

  Riemann zeta function - Wikipedia View original

  Is this image relevant?

• 

  File:Riemann zeta function absolute value.png - Wikimedia Commons View original

  Is this image relevant?

• 

  Euler product approximation of the Riemann zeta function - MathOverflow View original

  Is this image relevant?

• 

  Riemann zeta function - Wikipedia View original

  Is this image relevant?

1 of 3

• Riemann zeta function ζ(s) defined as infinite series $\sum_{n=1}^{\infty} \frac{1}{n^s}$ where s represents complex number
• Series converges absolutely for Re(s) > 1 establishing initial domain of convergence
• Series diverges for Re(s) ≤ 1 necessitating analytic continuation to extend function's domain
• Function expressed as Euler product over prime numbers $\zeta(s) = \prod_{p \text{ prime}} \frac{1}{1-p^{-s}}$
• Simple pole occurs at s = 1 with residue 1 relating to divergence of harmonic series
• Convergence domain analysis crucial for understanding function's behavior in different complex plane regions (critical strip, half-plane)

Key Properties and Significance

• Function plays central role in analytic number theory connecting complex analysis and prime number theory
• Serves as prototype for more general L-functions in number theory
• Encodes information about distribution of prime numbers through its analytic properties
• Integral to formulation of explicit formulas in prime number theory (prime counting function)
• Appears in various mathematical contexts (statistical mechanics, quantum chaos)
• Understanding zeta function properties essential for tackling Riemann Hypothesis and related conjectures (Generalized Riemann Hypothesis)

Analytic Continuation of the Zeta Function

Techniques and Process

• Analytic continuation extends ζ(s) definition beyond initial convergence domain to entire complex plane except s = 1
• Process utilizes complex analysis techniques (contour integration, functional equations)
• Riemann's functional equation provides means for continuation relating ζ(s) values in critical strip 0 < Re(s) < 1 to values outside
• Extended function maintains analytic properties of original function where they overlap ensuring unique well-defined continuation
• Reveals ζ(s) has trivial zeros at negative even integers and potentially non-trivial zeros in critical strip
• Techniques involve complex integration methods (Hankel contour)
• Mellin transform and inverse Mellin transform used in some approaches to analytic continuation

Importance and Applications

• Continuation essential for studying function's behavior in regions where original series definition diverges
• Allows exploration of function's properties throughout complex plane including critical strip
• Crucial for formulating and investigating Riemann Hypothesis
• Enables study of zeta function's special values and their number-theoretic significance
• Provides framework for analyzing other zeta and L-functions in number theory
• Techniques developed for zeta function continuation apply to broader class of Dirichlet series
• Understanding continuation process deepens insight into complex analysis and its applications in number theory

Functional Equation of the Zeta Function

Form and Properties

• Functional equation relates ζ(s) to ζ(1-s) providing symmetry in complex plane
• Complete form $\zeta(s) = 2^s \pi^{s-1} \sin(\frac{\pi s}{2}) \Gamma(1-s) \zeta(1-s)$ where Γ represents gamma function
• Allows computation of ζ(s) in critical strip 0 < Re(s) < 1 using values outside this region
• Implies ζ(s) has meromorphic continuation to entire complex plane with single pole at s = 1
• Reveals existence of trivial zeros at negative even integers corresponding to gamma function poles
• Demonstrates symmetry of non-trivial zeros about critical line Re(s) = 1/2
• Connects zeta function to other special functions (gamma function, sine function)

Applications and Significance

• Crucial for investigating zeta function properties across entire domain
• Enables efficient numerical computation of zeta function values
• Provides insight into location and distribution of zeta function zeros
• Used in proofs of various zeta function properties (functional equation for Dirichlet L-functions)
• Serves as model for functional equations of more general L-functions
• Plays role in formulating and studying Riemann Hypothesis
• Demonstrates deep connections between complex analysis and number theory

Zeros of the Zeta Function

Classification and Distribution

• Zeros classified into two types trivial zeros and non-trivial zeros
• Trivial zeros occur at negative even integers (-2, -4, -6, ...) consequence of functional equation
• Non-trivial zeros lie in critical strip 0 < Re(s) < 1 symmetric about real axis and critical line Re(s) = 1/2
• Riemann Hypothesis conjectures all non-trivial zeros lie on critical line
• Zero distribution closely related to prime number distribution expressed in explicit formulae of prime number theory
• Density of zeros near real axis described by Riemann-von Mangoldt formula providing asymptotic count of zeros up to given height
• First few non-trivial zeros approximately at 1/2 + 14.1347i, 1/2 + 21.0220i, 1/2 + 25.0109i (on critical line)

Significance and Applications

• Zero distribution crucial for applications in analytic number theory
• Provides improved estimates for prime counting function and gaps between primes
• Zeros encode information about fluctuations in prime number distribution
• Study of zero-free regions leads to best known error terms in Prime Number Theorem
• Connections to random matrix theory through statistical properties of zero spacings
• Investigation of zeros drives development of computational techniques in number theory
• Understanding zero distribution essential for progress on Riemann Hypothesis and related problems (Lindelöf Hypothesis)