Calculate Riemann Zeta Function

Online calculator and formulas for computing the Riemann Zeta Function - Key to prime number distribution

Riemann Zeta Function Calculator

Riemann Zeta Function

The ζ(s) or Riemann Zeta Function is one of the most important functions in analytic number theory and closely connected to prime number distribution.

Real number > 1 for convergence of the Zeta function
Result
ζ(s):

Riemann Zeta Function Curve

Mouse pointer on the graph shows the values.
At s = 1 the result is ∞. The Y-scale is limited to ±20 for better visualization.

Formulas for the Riemann Zeta Function

Dirichlet Series
\[\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}\]

For Re(s) > 1

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

Product over all primes p

Integral Representation
\[\zeta(s) = \frac{1}{\Gamma(s)} \int_0^{\infty} \frac{x^{s-1}}{e^x-1} dx\]

For Re(s) > 1

Functional Equation
\[\zeta(s) = 2^s \pi^{s-1} \sin\left(\frac{\pi s}{2}\right) \Gamma(1-s) \zeta(1-s)\]

Riemann functional equation

Analytic Continuation
\[\zeta(s) = \frac{1}{s-1} + \gamma - \frac{\gamma_1}{1!}(s-1) + \frac{\gamma_2}{2!}(s-1)^2 + ...\]

Laurent series around s = 1 with Stieltjes constants γₙ

Special Values

Famous Values
ζ(2) = π²/6 ζ(4) = π⁴/90 ζ(6) = π⁶/945 ζ(0) = -1/2
Prime Numbers
Connection to primes

Euler product shows deep connection to prime number theory

Pole at s = 1
\[\zeta(1) = \infty\]

Simple pole with residue 1

Applications

Prime number theory, analytic number theory, mathematical physics and cryptography.

The Riemann Hypothesis

One of the most famous unsolved problems in mathematics:

"All nontrivial zeros of the Riemann zeta function have real part 1/2"

Millennium Problem: One of the seven Millennium Problems with a prize of 1 million US dollars. The hypothesis has profound implications for the distribution of prime numbers.

Detailed Description of the Riemann Zeta Function

Mathematical Definition

The Riemann Zeta Function is one of the central functions in analytic number theory. It was systematically studied by Bernhard Riemann and reveals a deep connection between prime numbers and analysis.

Definition: ζ(s) = Σ(n=1 to ∞) 1/n^s for Re(s) > 1
Using the Calculator

Enter the argument s and click 'Calculate'. For s = 1 the function is undefined (pole). The series converges for Re(s) > 1.

Historical Background

The function was originally introduced by Leonhard Euler as a Dirichlet series. Bernhard Riemann recognized its fundamental importance for prime number theory and proved the famous functional equation.

Properties and Applications

Mathematical Applications
  • Prime number theorem and prime distribution
  • Analytic number theory (L-functions)
  • Additive combinatorics
  • Modular forms and automorphic functions
Physical Applications
  • Quantum chaos and spectral theory
  • Statistical mechanics (partition functions)
  • String theory and quantum field theory
  • Critical phenomena and phase transitions
Special Properties
  • Euler Product: Connection to prime numbers
  • Functional Equation: Symmetry around s = 1/2
  • Zeros: Triviale (-2, -4, -6, ...) and nontrivial
  • Pol: Einfacher Pol bei s = 1 mit Residuum 1
Interesting Facts
  • ζ(2) = π²/6 löst das Basler Problem
  • ζ(-1) = -1/12 (Ramanujan-Summe)
  • Die ersten 10¹³ nichttrivialen Nullstellen liegen auf der kritischen Linie
  • Verbindung zu Random Matrix Theory

Berechnungsbeispiele

Beispiel 1

ζ(2) = π²/6 ≈ 1.6449

Lösung des Basler Problems (Euler, 1734)

Beispiel 2

ζ(4) = π⁴/90 ≈ 1.0823

Weitere exakte Lösung von Euler

Beispiel 3

ζ(0) = -1/2

Analytische Fortsetzung

Verbindung zur Primzahlentheorie

Euler-Produkt

Das Euler-Produkt zeigt die fundamentale Verbindung zu Primzahlen:

\[\zeta(s) = \prod_{p \text{ prim}} \frac{1}{1-p^{-s}}\]

Diese Darstellung ist der Schlüssel zum Verständnis der Primzahlenverteilung.

Primzahlsatz

Die Nullstellen der Zeta-Funktion bestimmen die Genauigkeit des Primzahlsatzes:

\[\pi(x) \sim \frac{x}{\ln x}\]

Wobei π(x) die Anzahl der Primzahlen ≤ x ist.

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