# 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:

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

### Critical Strip

Let $s = \sigma + i t$.

The region defined by the equation $0 < \sigma < 1$ is known as the **critical strip**.

### Critical Line

Let $s = \sigma + i t$.

The line defined by the equation $\sigma = \dfrac 1 2$ is known as the **critical line**.

Hence the popular form of the statement of the Riemann Hypothesis:

*All the nontrivial zeroes of the Riemann zeta function lie on the***critical line**.

## 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 $\zeta$.

## Zeroes

The **zeroes** of the Riemann zeta function are, according to the conventional definition of the zero of a (complex) function, the points $s \in \C$ such that:

- $\map \zeta s = 0$

### Trivial Zeroes

The **trivial zeroes** of the Riemann zeta function $\zeta$ are the strictly negative even integers :

- $\set {n \in \Z: n = -2 \times k: k \in \N_{\ne 0} } = \set {-2, -4, -6, \ldots}$

### Nontrivial Zeroes

The **nontrivial zeroes** of the Riemann zeta function $\zeta$ are the zeroes of $\zeta$ that are not trivial.

## 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 are Even Negative Integers

### 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 ($\text {1826}$ – $\text {1866}$) - 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): $0 \cdotp 5$ - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**zeta function** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**zeta function** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**zeta function**