Riemann Zeta Function

📐 Definition

For complex numbers $s$ with $\Re(s) > 1$ , the Riemann zeta function is defined by the absolutely convergent Dirichlet series

\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}

This function admits a unique analytic continuation to the complex plane, except for a simple pole at $s = 1$ .

⚙️ Key Properties

Euler Product (Connection to Primes)

For $\Re(s) > 1$ , the zeta function factorizes into an infinite product over the prime numbers:

\zeta(s) = \prod_{p \text{ prime}} \frac{1}{1 - p^{-s}}

This identity establishes the fundamental link between $\zeta(s)$ and the distribution of primes.

Analytic Continuation

The zeta function extends analytically to all $s \in \mathbb{C}\setminus \{1\}$ . One convenient representation (valid for $\Re(s)>1$ ) is

\zeta(s) = \frac{1}{\Gamma(s)} \int_{0}^{\infty} \frac{x^{s-1}}{e^{x} - 1}\, dx

since $\int_0^\infty \frac{x^{s-1}}{e^{x}-1}\,dx = \Gamma(s)\zeta(s)$ in that half-plane.

Functional Equation

The zeta function satisfies a deep symmetry relating values at $s$ and $1 - s$ :

\zeta(s) = 2^s \pi^{s-1} \sin\!\left(\frac{\pi s}{2}\right) \Gamma(1-s)\, \zeta(1-s)

This functional equation underlies the symmetry of the nontrivial zeros about the critical line $\Re(s) = \tfrac12$ .

Pole at $s = 1$

The zeta function has a single simple pole at $s = 1$ , with residue $1$ :

\zeta(s) = \frac{1}{s-1} + \gamma + \mathcal{O}(s-1)

where $\gamma$ is the Euler–Mascheroni constant.

Trivial Zeros

The zeta function vanishes at the negative even integers:

\zeta(-2n) = 0, \qquad n = 1, 2, 3, \dots

These are known as the trivial zeros.

Asymptotic Behavior

As $|s| \to \infty$ in vertical strips, the zeta function exhibits controlled growth. In particular,

\zeta(s) = 1 + \mathcal{O}\!\left(2^{-\Re(s)}\right) \qquad (\Re(s) \to +\infty)

More refined bounds depend on $\Im(s)$ and are central in analytic number theory.

Zeros in the Complex Plane

The remaining zeros of $\zeta(s)$ , known as nontrivial zeros, lie in the critical strip

0 < \Re(s) < 1

They occur symmetrically with respect to the real axis and the critical line $\Re(s) = \tfrac12$ .

The Riemann Hypothesis asserts that all nontrivial zeros satisfy $\Re(s) = \tfrac12$ .

🎯 Special Values

At selected arguments, the zeta function takes notable exact or well-known values:

$s$	$\zeta(s)$ (exact)	Approximate value
$2$	$\tfrac{\pi^2}{6}$	$1.6449340668$
$4$	$\tfrac{\pi^4}{90}$	$1.0823232337$
$0$	$-\tfrac12$	$-0.5$
$-1$	$-\tfrac{1}{12}$	$-0.0833333333$
$-3$	$\tfrac{1}{120}$	$0.0083333333$

Gamma Function Appears in the functional equation and integral representation

Polygamma Functions Related via ψ^(n)(z) = (-1)^(n+1) n! ζ(n+1, z)

Digamma Function Pole residue involves the Euler-Mascheroni constant

Usage in Oakfield

Historical Foundations

📜 Early Foundations (17th–18th Centuries)

The study of sums of reciprocal powers began with Pierre de Fermat and Jakob Bernoulli. Leonhard Euler (1730s) made the decisive breakthrough by evaluating $\zeta(2)$ , $\zeta(4)$ , and related values, and by discovering the Euler product linking $\zeta(s)$ to prime numbers.
🔬 Riemann’s Contribution (19th Century)

Bernhard Riemann’s 1859 memoir introduced the analytic continuation, functional equation, and the study of complex zeros. This work laid the foundation of modern analytic number theory.
🌍 Modern Developments

The zeta function now plays a central role across mathematics and physics, from prime number theory and random matrix theory to quantum chaos and statistical mechanics. Computational verification of its zeros has driven advances in numerical analysis and high-performance computing.

📚 References

B. Riemann, Über die Anzahl der Primzahlen unter einer gegebenen Grösse (1859)
NIST Digital Library of Mathematical Functions (DLMF), §25
Titchmarsh, The Theory of the Riemann Zeta-Function