Riemann zeta function

In mathematics, the Riemann zeta function, named after Bernhard Riemann, is a function of significant importance in number theory, because of its relation to the distribution of prime numbers. It also has applications in other areas such as physics, probability theory, and applied statistics.

Definition

The Riemann zeta-function ζ(s) is defined for any complex number s with real part > 1 by the Dirichlet series:

\zeta(s) = \sum_{n=1}^\infin \frac{1}{n^s} In the region {s ∈ C: Re(s) > 1}, this infinite series converges and defines a function analytic in this region. Bernhard Riemann realized that the zeta-function can be extended by analytic continuation in a unique way to a meromorphic function ζ(s) defined for all complex numbers s with s ≠ 1. It is this function that is the object of the Riemann hypothesis.

Relationship to prime numbers

The connection between this function and prime numbers was already realized by Leonhard Euler:

\zeta(s) = \prod_{p\in\mathbb{P}} \frac{1}{1-p^{-s}} an infinite product extending over all prime numbers p. This is called an Euler product formula and converges for Re(s) > 1. It is a consequence of two simple and fundamental results in mathematics; the formula for the geometric series and the fundamental theorem of arithmetic. The formula is proved here.

Various properties

For the Riemann zeta function on the critical line, see Z-function. For sums involving the zeta-function at integer values, see rational zeta series.

Values at the integers

The following are values of the zeta function for the first few natural numbers.

\zeta(1) = 1 + \frac{1}{2} + \frac{1}{3} + \cdots = \infty

; this is the harmonic series.

\zeta(2) = 1 + \frac{1}{2^2} + \frac{1}{3^2} + \cdots = \frac{\pi^2}{6} \approx 1.645

; the demonstration of this equality is known as the Basel problem.

\zeta(3) = 1 + \frac{1}{2^3} + \frac{1}{3^3} + \cdots \approx 1.202

; this is called Apéry's constant

\zeta(4) = 1 + \frac{1}{2^4} + \frac{1}{3^4} + \cdots = \frac{\pi^4}{90} \approx 1.0823

Zeros of the Riemann zeta function

The Riemann zeta function has zeros at the negative even integers. These are called the trivial zeros. It is known that any non-trivial zero lies in the open strip {s ∈ C: 0 < Re(s) < 1}. The Riemann hypothesis asserts that any non-trivial zero s has Re(s) = 1/2. In the theory of the Riemann zeta function, {s ∈ C: Re(s) = 1/2} is called the critical line.

The zeros of ζ(s) are important because certain line integrals involving the function ln(1/ζ(s)) can be used to approximate the prime counting function π(x). These path integrals are computed with the residue theorem and hence knowledge of the integrand's singularities is required.

The functional equation

The zeta-function satisfies the following functional equation:

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

valid for all s in C\{0,1}. Here, Γ denotes the gamma function. This formula is used to construct the analytic continuation in the first place. At s = 1, the zeta-function has a simple pole with residue 1.

There is also a symmetric version of the functional equation, given by first defining

\xi(s) = \pi^{-s/2}\Gamma\left(\frac{s}{2}\right)\zeta(s).

The functional equation is then given by

\xi(s) = \xi(1 - s).

The reciprocal

The reciprocal of the zeta function may be expressed as a Dirichlet series over the Möbius function μ(n):

\frac{1}{\zeta(s)} = \sum_{n=1}^{\infin} \frac{\mu(n)}{n^s} for every complex number s with real part > 1. There are a number of similar relations involving various well-known multiplicative functions; these are given in the article on the Dirichlet series.

The above, together with the expression for ζ(2), can be used to prove that the probability of two random integers being coprime is 6/π². The Riemann hypothesis is equivalent to the claim that this expression is valid when the real part of s is greater than 1/2.

The Riemann zeta function as a Mellin transform

The Mellin transform of a function f(x) is defined as

\{ \mathcal{M} f \}(s) = \int_0^\infty f(x)x^s \frac{dx}{x}

in the region where the integral is defined. There are various expressions for the zeta-function as a Mellin transform. If the real part of s is greater than one, we have

\Gamma(s)\zeta(s) =\left\{ \mathcal{M} \left(\frac{1}{\exp(x)-1}\right) \right\}(s)

By subtracting off the first terms of the power series expansion of 1/(exp(x) − 1) around zero, we can get the zeta-function in other regions. In particular, in the critical strip we have

\Gamma(s)\zeta(s) = \left\{ \mathcal{M}\left(\frac{1}{\exp(x)-1}-\frac1x\right)\right\}(s)

and when the real part of s is between −1 and 0,

\Gamma(s)\zeta(s) = \left\{\mathcal{M}\left(\frac{1}{\exp(x)-1}-\frac1x+\frac12\right)\right\}(s)

We can also find expressions which relate to prime numbers and the prime number theorem. If π(x) is the prime counting function, then

\log \zeta(s) = s \int_0^\infty \frac{\pi(x)}{x(x^s-1)}dx

for values with $\Re(s)>1$ . We can relate this to the Mellin transform of π(x) by $\frac{\log \zeta(s)}{s} - \omega(s) = \left\{\mathcal{M} \pi(x)\right\}(-s)$ where

\omega(s) = \int_0^\infty \frac{\pi(s)}{x^{s+1}(x^s-1)}dx

converges for $\Re(s)>\frac12$ .

A similar Mellin transform involves the Riemann prime counting function J(x), which counts prime powers pⁿ with a weight of 1/n, so that $J(x) = \sum \frac{\pi(x^{1/n})}{n}.$ Now we have

\frac{\log \zeta(s)}{s} = \left\{\mathcal{M} J \right\}(-s)

These expressions can be used to prove the prime number theorem by means of the inverse Mellin transform. Riemann's prime counting function is easier to work with, and π(x) can be recovered from it by Möbius inversion.

Series expansions

The Riemann zeta function is meromorphic with a single pole of order one at s = 1. It can therefore be expanded as a Laurent series about s = 1; the series development then is

\zeta(s) = \frac{1}{s-1} + \gamma_0 + \gamma_1(s-1) + \gamma_2(s-1)^2 + \cdots.

The constants here are called the Stieltjes constants and can be defined as

\gamma_k = \frac{(-1)^k}{k!} \lim_{N \rightarrow \infty} \left(\sum_{m \le N} \frac{\ln^k m}{m} - \frac{\ln^{k+1}N}{k+1}\right).

The constant term γ₀ is the Euler-Mascheroni constant.

Another series development valid for the entire complex plane is

\zeta(s) = \frac{1}{s-1} - \sum_{n=1}^\infty (\zeta(s+n)-1)\frac{s^{\overline{n}}}{(n+1)!}

where $s^{\overline{n}}$ is the rising factorial $s^{\overline{n}} = s(s+1)\cdots(s+n-1)$ . This can be used recursively to extend the Dirichlet series definition to all complex numbers.

The Riemann zeta function also appears in a form similar to the Mellin transform in an integral over the Gauss-Kuzmin-Wirsing operator acting on x^s−1; that context gives rise to a series expansion in terms of the falling factorial.

Hadamard product

The infinite product expansion of the Riemann zeta function over the non-trivial zeros $\rho$ is due to Hadamard:

\zeta(s) = \frac{e^{As}}{2(s-1)\Gamma(1+s/2)} \prod_\rho \left(1 - \frac{s}{\rho} \right) e^{s/\rho}

where $A = \log(2 \pi) - 1 - \gamma/2$ , and the term $\gamma$ is again the Euler-Mascheroni constant.

Globally convergent series

A globally convergent series for the zeta function, valid for all complex numbers s except s = 1, was conjectured by Konrad Knopp and proved by Helmut Hasse in 1930:

\zeta(s)=\frac{1}{1-2^{1-s}}

\sum_{n=0}^\infty \frac {1}{2^{n+1}} \sum_{k=0}^n (-1)^k {n \choose k} (k+1)^{-s}.

The series only appeared in an Appendix to Hasse's paper, and did not become generally known until it was rediscovered more than 60 years later (see Sondow, 1994).

Peter Borwein has shown a very rapidly convergent series suitable for high precision numerical calculations. The algorithm, making use of Chebyshev polynomials, is described in the article on the Dirichlet eta function.

Universality

The critical strip of the Riemann zeta function has the remarkable property of universality. This zeta-function universality states that there exists some location on the critical strip that approximates any holomorphic function arbitrarily well. Since holomorphic functions are very general, this property is quite remarkable.

Applications

Although mathematicians regard the Riemann zeta function as being primarily relevant to the "purest" of mathematical disciplines, number theory, it also occurs in applied statistics (see Zipf's law and Zipf-Mandelbrot law), physics, and the mathematical theory of musical tuning.

During several physics-related calculations, one must evaluate the sum of the positive integers; paradoxically, on physical grounds one expects a finite answer. When this situation arises, there is typically a rigorous approach involving much in-depth analysis, as well as a "short-cut" solution relying on the Riemann zeta-function. The argument goes as follows: we wish to evaluate the sum $1 + 2 + 3 + \cdots$ , but we can re-write it as a sum of reciprocals:

$S\,\!$	$=1 + 2 + 3 + 4 + \cdots$
	$=\left(\frac{1}{1}\right)^{-1} + \left(\frac{1}{2}\right)^{-1} + \left(\frac{1}{3}\right)^{-1} + \left(\frac{1}{4}\right)^{-1} + \cdots$
	$=\sum_{n=1}^{\infin} \frac{1}{n^{-1}}.$

The sum S appears to take the form of $\zeta(-1)$ . However, −1 lies outside of the domain for which the Dirichlet series for the zeta-function converges. However, a divergent series of positive terms such as this one can sometimes be summed in a reasonable way by the method of Ramanujan summation (see Hardy, Divergent Series.) Ramanujan summation involves an application of the Euler-Maclaurin summation formula, and when applied to the zeta-function, it extends its definition to the whole complex plane. In particular

1+2+3+\cdots = -\frac{1}{12} (\Re)

where the notation $(\Re)$ indicates Ramanujan summation.

For even powers we have:

1+2^{2k}+3^{2k}+\cdots = 0 (\Re)

and for odd powers we have a relation with the Bernoulli numbers:

1+2^{2k+1}+3^{2k+1}+\cdots = -\frac{B_{2k}}{2k} (\Re)

Zeta function regularization is used as one possible means of regularization of divergent series in quantum field theory. In one notable example, the Riemann zeta-function shows up explicitly in the calculation of the Casimir effect.

Generalizations

There are a number of related zeta functions that can be considered to be generalizations of Riemann's zeta-function. The simplest of these are the Hurwitz zeta function

\zeta(s,q) = \sum_{k=0}^\infty (k+q)^{-s},

which coincides with Riemann's zeta-function when q = 1.

The polylogarithm is given by

Li_s(z) \equiv \sum_{k=1}^\infty {z^k \over k^s}

which coincides with Riemann's zeta-function when z = 1.

The Lerch transcendent is given by

\Phi(z, s, q) = \sum_{k=0}^\infty

\frac { z^k} {(k+q)^s} which coincides with Riemann's zeta-function when z = 1 and q = 1.

The Clausen function $Cl_{s} ( \theta )$ that can be chosen as the Real or Imaginary part of $Li_{s} (e^{i\theta})$

Zeta-functions in fiction

Neal Stephenson's 1999 novel Cryptonomicon mentions the zeta-function as a pseudo-random number source, a useful component in cipher design.

References

Bernhard Riemann, Über die Anzahl der Primzahlen unter einer gegebenen Grösse (1859). In Gesammelte Werke, Teubner, Leipzig (1892), Reprinted by Dover, New York (1953).
Jacques Hadamard, Sur la distribution des zéros de la fonction ζ(s) et ses conséquences arithmétiques, Bulletin de la Societé Mathématique de France 14 (1896) pp 199-220.
Helmut Hasse, Ein Summierungsverfahren für die Riemannsche ζ-Reihe, (1930) Math. Z. 32 pp 458-464. (Globally convergent series expression.)
(links to PDF file)
Jonathan Sondow, "Analytic continuation of Riemann's zeta function and values at negative integers via Euler's transformation of series", Proc. Amer. Math. Soc. 120 (1994) 421-424.

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