7.3 Special values and identities involving the zeta function

The Riemann zeta function is a powerhouse in number theory, with special values that reveal deep mathematical truths. From the Basel problem to Apéry's constant, these values connect seemingly unrelated concepts and offer glimpses into the function's profound nature.

Functional equations and identities involving the zeta function unlock its secrets, tying it to prime numbers and complex analysis. The Riemann Hypothesis, if proven, would revolutionize our understanding of prime distribution and reshape multiple mathematical fields.

Special Values

Bernoulli Numbers and the Euler-Mascheroni Constant

• Bernoulli numbers form a sequence of rational numbers with deep connections to number theory and analysis
• Defined recursively through the generating function $\frac{x}{e^x - 1} = \sum_{n=0}^{\infty} B_n \frac{x^n}{n!}$
• First few Bernoulli numbers include B₀ = 1, B₁ = -1/2, B₂ = 1/6, B₄ = -1/30
• Play crucial roles in expressing sums of powers and in the values of the Riemann zeta function at even integers
• Euler-Mascheroni constant, denoted by γ, emerges as the limit of the difference between the harmonic series and the natural logarithm
• Defined mathematically as $\gamma = \lim_{n \to \infty} \left(\sum_{k=1}^n \frac{1}{k} - \ln n\right)$
• Approximately equal to 0.57721566490153286060651209008240243104215933593992...
• Appears in various mathematical contexts, including the study of the Riemann zeta function and prime number theory

The Basel Problem and Apéry's Constant

• Basel problem involves finding the exact sum of the reciprocals of the squares of positive integers
• Solved by Euler in 1735, demonstrating that $\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}$
• Solution connects the Riemann zeta function to π, as ζ(2) = π²/6
• Generalizes to other even powers, with ζ(2n) expressible in terms of π and Bernoulli numbers
• Apéry's constant, denoted by ζ(3), represents the sum of the reciprocals of the cubes of positive integers
• Irrational number approximately equal to 1.2020569031595942853997381615114499907649862923404988...
• Proven irrational by Roger Apéry in 1979, marking a significant advancement in number theory
• Appears in various physical and mathematical contexts, including quantum electrodynamics and knot theory

Functional Equations and Identities

The Functional Equation of the Riemann Zeta Function

• Riemann zeta function satisfies a fundamental functional equation relating its values at s and 1-s
• Expressed as $\zeta(s) = 2^s \pi^{s-1} \sin\left(\frac{\pi s}{2}\right) \Gamma(1-s) \zeta(1-s)$
• Allows extension of the zeta function to the entire complex plane, except for a simple pole at s = 1
• Reveals symmetry in the behavior of the zeta function across the critical line Re(s) = 1/2
• Provides crucial insights into the distribution of zeros of the zeta function
• Utilized in various proofs and investigations related to the Riemann hypothesis

The Riemann Hypothesis and Its Implications

• Riemann hypothesis states that all non-trivial zeros of the zeta function have real part equal to 1/2
• Considered one of the most important unsolved problems in mathematics
• Equivalent to stating that all zeros of the zeta function in the critical strip 0 < Re(s) < 1 lie on the critical line Re(s) = 1/2
• Has profound implications for the distribution of prime numbers
• Connects to various areas of mathematics, including complex analysis, number theory, and probability theory
• Verification of the hypothesis would lead to improved error terms in many number-theoretic estimates
• Numerous equivalent formulations and generalizations exist, highlighting its fundamental nature

The Prime Number Theorem and Zeta Function Connections

• Prime Number Theorem describes the asymptotic distribution of prime numbers
• States that the number of primes less than or equal to x, denoted π(x), is asymptotically equivalent to x/ln(x)
• Can be expressed in terms of the Riemann zeta function as $\pi(x) \sim \text{Li}(x)$, where Li(x) is the logarithmic integral
• Proof relies heavily on complex analysis and properties of the Riemann zeta function
• Demonstrates deep connection between analytic properties of the zeta function and arithmetic properties of prime numbers
• Provides error estimates for the distribution of primes, with improvements contingent on the Riemann hypothesis
• Generalizations exist for arithmetic progressions and number fields, leading to broader applications in number theory

Generalizations

L-functions and Their Properties

• L-functions generalize the Riemann zeta function to broader arithmetic contexts
• Include Dirichlet L-functions associated with Dirichlet characters, modular L-functions, and automorphic L-functions
• Typically defined as Dirichlet series with an Euler product representation
• Satisfy functional equations analogous to that of the Riemann zeta function
• Exhibit analytic continuation to the entire complex plane (with possible exceptions at s = 0 and s = 1)
• Play crucial roles in various number-theoretic problems, including the study of prime numbers in arithmetic progressions
• Dirichlet L-functions used to prove Dirichlet's theorem on primes in arithmetic progressions
• Modular L-functions connected to elliptic curves and the proof of Fermat's Last Theorem
• Generalized Riemann hypothesis extends to L-functions, positing that their non-trivial zeros lie on the critical line
• Provide a unifying framework for studying various arithmetic objects and their associated zeta functions