# Definition:Riemann Zeta Function

## Contents

## Definition

The **Riemann Zeta Function** $\zeta$ is the complex function defined on the half-plane $\map \Re s > 1$ as the series:

- $\displaystyle \map \zeta s = \sum_{n \mathop = 1}^\infty \frac 1 {n^s}$

## Analytic Continuation

By Analytic Continuations of Riemann Zeta Function, $\zeta$ has a unique analytic continuation to $\C \setminus \set 1$.

This analytic continuation is still called the **Riemann zeta function** and still denoted $\zeta$.

## Also see

- Results about
**the Riemann $\zeta$ function**can be found here.

### Special values

- Basel Problem, for $\zeta(2)$.
- Riemann Zeta Function at Even Integers
- Riemann Zeta Function at Non-Positive Integers
- Harmonic Series is Divergent: $\map \zeta s \to +\infty$ as $s \to 1$
- Trivial Zeroes of Riemann Zeta Function

### Generalizations

## Source of Name

This entry was named for Georg Friedrich Bernhard Riemann.

## Historical Note

The Riemann zeta function was discussed by Bernhard Riemann in his $1859$ article *Ueber die Anzahl der Primzahlen under einer gegebenen Grösse*.

In that paper he made several statements about that function.

Studying it led Jacques Salomon Hadamard to his proof of the Prime Number Theorem.

All of the statements made in that paper have now been proved except for one.

The last remaining statement which has not been resolved is the Riemann Hypothesis.

When Riemann first investigated this function, he used $s$ instead of the more typical complex variable $z$.

To this day, $s = \sigma + i t$ is still typically used as the complex variable in investigations of the zeta function.

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $0 \cdotp 5$ - 1992: Larry C. Andrews:
*Special Functions of Mathematics for Engineers*... (previous) ... (next): $\S 1.2.2$: Summary of convergence tests (footnote) - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.32$: Riemann ($1826$ – $1866$) - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {B}.19$: The Series $\sum 1/ p_n$ of the Reciprocals of the Primes - 1997: Donald E. Knuth:
*The Art of Computer Programming: Volume 1: Fundamental Algorithms*(3rd ed.) ... (previous) ... (next): $\S 1.2.7$: Harmonic Numbers: $(5)$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $0 \cdotp 5$