Definition:Riemann Zeta Function
Definition
The Riemann $\zeta$ (zeta) function is the complex function defined on the half-plane $\map \Re s > 1$ as the series:
- $\ds \map \zeta s = \sum_{n \mathop = 1}^\infty \frac 1 {n^s}$
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 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 are the zeroes of $\zeta$ that are not trivial.
Also known as
The Riemann zeta function is also often known as just the zeta function.
Some sources use hyphens: Riemann zeta-function or zeta-function
Some sources style it as Riemann's zeta function.
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
Investigation of what is now known as the Riemann zeta function goes back as far as Leonhard Paul Euler, who showed in $1744$ that $\ds \sum_{n \mathop = 1}^\infty \frac 1 {n^s} = \prod_{\text {$p$ prime} } \frac 1 {1 - p^{-s} }$.
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
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 35$: Miscellaneous Special Functions: Riemann Zeta Function $\map \zeta x = \dfrac 1 {1^x} + \dfrac 1 {2^x} + \dfrac 1 {3^x} + \cdots$
- 1985: Aleksandar Ivić: The Riemann Zeta-Function ... (previous) ... (next): Chapter $1$. Elementary Theory: $1.1$ Definition of $\map \zeta s$ and Elementary Properties
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $0 \cdotp 5$
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): zeta function
- 1992: Larry C. Andrews: Special Functions of Mathematics for Engineers (2nd ed.) ... (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): Riemann zeta function
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Riemann zeta function
- 2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 36$: Miscellaneous and Riemann Zeta Functions: Riemann Zeta Function $\map \zeta x = \dfrac 1 {1^x} + \dfrac 1 {2^x} + \dfrac 1 {3^x} + \cdots$
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): zeta function
- 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): zeta function