Riemann zeta function

In mathematics, the Riemann zeta function is a function which is of paramount importance in number theory, because of its relation to the distribution of prime numbers. It also has applications in other areas such as physics.

 Contents

Definition

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Riemann zeta function for real s>1

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

[itex]

\zeta(s) = \sum_{n=1}^\infin \frac{1}{n^s} [itex] In the region {s in 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:

[itex]

\zeta(s) = \prod_{p} \frac{1}{1-p^{-s}} [itex] an infinite product extending over all prime numbers p. This is called a Euler product formula and is a consequence of the two simple and fundamental results in mathematics; the formula for the geometric series and the fundamental theorem of arithmetic.

Proving the Euler product formula

Each term (for a given prime p) in the product above can be expanded to a geometric series consisting of the reciprocal of p raised to multiples of s, as follows

[itex]\frac{1}{1-p^{-s}} = 1 + \frac{1}{p^s} + \frac{1}{p^{2s}} + \frac{1}{p^{3s}} + \cdots + \frac{1}{p^{ks}} + \cdots [itex]

When [itex]\Re(s)>1[itex], [itex]\left|p^{-s}\right| < 1[itex] and this series converges absolutely. Hence we may take a finite number of factors, multiply them together, and rearrange terms. Taking all the primes p up to some prime number limit q, we have

[itex]\left|\zeta(s) - \prod_{p \le q}\left(1-\frac{1}{p^s}\right)^{-1}\right| < \sum_{n=q+1}^\infty \frac{1}{n^\sigma}[itex]

where σ is the real part of s. By the fundamental theorem of arithmetic, the partial product when expanded out gives a sum consisting of those terms [itex]\frac{1}{n^s}[itex] where n is a product of primes less than or equal to q. The inequality results from the fact that therefore only integers larger than q can fail to appear in this expanded out partial product. Since the difference between the partial product and ζ(s) goes to zero when σ > 1, we have convergence in this region.

An easier proof (for the layperson)

This proof only makes use of simple algebra that most high school students could understand. There is a certain sieving property that we can use to our advantage:

[itex]\zeta(s) = 1+\frac{1}{2^s}+\frac{1}{3^s}+\frac{1}{4^s}+\frac{1}{5^s}+ \cdots [itex]
[itex]\frac{1}{2^s}\zeta(s) = \frac{1}{2^s}+\frac{1}{4^s}+\frac{1}{6^s}+\frac{1}{8^s}+\frac{1}{10^s}+ \cdots [itex]

Subtracting the second from the first we remove all elements that have a factor of 2:

[itex]\left(1-\frac{1}{2^s}\right)\zeta(s) = 1+\frac{1}{3^s}+\frac{1}{5^s}+\frac{1}{7^s}+\frac{1}{9^s}+ \cdots [itex]

Repeating for the next term:

[itex]\left(1-\frac{1}{2^s}\right)\zeta(s) = 1+\frac{1}{3^s}+\frac{1}{5^s}+\frac{1}{7^s}+\frac{1}{9^s}+ \cdots [itex]
[itex]\frac{1}{3^s}\left(1-\frac{1}{2^s}\right)\zeta(s) = \frac{1}{3^s}+\frac{1}{9^s}+\frac{1}{15^s}+\frac{1}{21^s}+\frac{1}{27^s}+ \cdots [itex]

Subtracting again we get:

[itex]\left(1-\frac{1}{3^s}\right)\left(1-\frac{1}{2^s}\right)\zeta(s) = 1+\frac{1}{5^s}+\frac{1}{7^s}+\frac{1}{11^s}+\frac{1}{13^s}+ \cdots [itex]

It can be seen that the right side is being sieved. Repeating infinitely we get:

[itex] \cdots \left(1-\frac{1}{11^s}\right)\left(1-\frac{1}{7^s}\right)\left(1-\frac{1}{5^s}\right)\left(1-\frac{1}{3^s}\right)\left(1-\frac{1}{2^s}\right)\zeta(s) = 1 [itex]

Dividing both sides by everything but the [itex]\zeta(s)[itex] we get:

[itex] \zeta(s) = \frac{1}{\left(1-\frac{1}{2^s}\right)\left(1-\frac{1}{3^s}\right)\left(1-\frac{1}{5^s}\right)\left(1-\frac{1}{7^s}\right)\left(1-\frac{1}{11^s}\right) \cdots } [itex]

This can be written more concisely as an infinite product over all primes p :

[itex]\zeta(s)\;=\;\prod_{p} (1-p^{-s})^{-1}.[itex]

The importance of the zeros of ζ(s)

The zeros of ζ(s) are important because certain path 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.

Basic properties

The zeta function satisfies the following functional equation:

[itex]

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

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

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

The functional equation is then given by

[itex]\xi(s) = \xi(1 - s).[itex]

Euler was also able to calculate ζ(2k) for even integers 2k using the formula

[itex]

\zeta(2k) = (-1)^{k+1}\frac{B_{2k}(2\pi)^{2k}}{2(2k)!} [itex]

where B2k are the Bernoulli numbers. From this one sees that ζ(2) = π2/6, ζ(4) = π4/90, ζ(6) = π6/945 etc. (sequence A046988 (http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A046988)/A002432 (http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A002432) in OEIS). These give well-known infinite series for π. For odd integers the case is not so simple; Apry proved that ζ(3) was an irrational number, and using related methods it can be shown an infinite number of other ζ values at odd positive integers are irrational. For a discussion of odd negative values, see Bernoulli numbers. For the zeta function on the critical line, see Z function.

One can express the reciprocal of the zeta function using the Mbius function μ(n) as follows:

[itex]

\frac{1}{\zeta(s)} = \sum_{n=1}^{\infin} \frac{\mu(n)}{n^s} [itex] for every complex number s with real part > 1. This, together with the above 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

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

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

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

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

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

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

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

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

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

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

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

converges for [itex]\Re(s)>\frac12[itex].

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 [itex]J(x) = \sum \frac{\pi(x^{1/n})}{n}.[itex] Now we have

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

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 Mbius 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

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

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

[itex]\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).[itex]

The constant term γ0 is the Euler-Mascheroni constant.

Another series development valid for the entire complex plane is

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

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

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

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 [itex]1 + 2 + 3 + \cdots[itex], but we can re-write it as a sum of reciprocals:

 [itex]S\,\![itex] [itex]=1 + 2 + 3 + 4 + \cdots[itex] [itex]=\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[itex] [itex]=\sum_{n=1}^{\infin} \frac{1}{n^{-1}}.[itex]

The sum S appears to take the form of [itex]\zeta(-1)[itex]. However, −1 lies outside of the domain for which the Dirichlet series for zeta 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

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

the notation indicating Ramanujan summation.

This problem arises in the Casimir effect, where infinitely many contributions must add to produce a finite (and experimentally small) force. Likewise, when applying quantum mechanics to the relativistic string, equations arise containing operators which must be placed in the proper order. The situation is much like that encountered in introductory quantum mechanics, where one meets mathematical quantities that do not commute: [itex]AB \neq BA[itex]. If, for example, [itex]AB - BA = 1[itex], we can exchange the order of the operators [itex]A[itex] and [itex]B[itex], but at the cost of adding a constant. Quantizing the relativistic string requires this performance to be conducted infinitely many times, requiring infinitely many constants—in fact, the sum of all positive integers.

Generalizations

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

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

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

The polylogarithm is given by

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

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

The Lerch transcendant is given by

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

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

Zeta functions in fiction

Neal Stephenson's 1999 novel Cryptonomicon mentions [itex]\zeta(z)[itex] as a pseudo-random number source, a useful component in cipher design.

References

• Edwards, H. M., Riemann's Zeta Function, Academic Press, 1974. ISBN 0-486-41740-9
• Ivic, A., The Riemann Zeta Function, John Wiley & Sons, 1985. ISBN 0-471-80634-X
• Titchmarsh, E. C., The Theory of the Riemann Zeta Function, second revised (Heath-Brown) edition, Oxford University Press, 1986
• Hardy, G.H., Divergent Series, Oxford, Clarendon Press, 1949de:Zeta-Funktion

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