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 physics.
| Table of contents |
|
2 Relationship to prime numbers 3 Basic properties 4 Applications 5 Zeta Functions in Fiction |
The Riemann zeta function ζ(s) is defined for any complex number s with real part > 1 as:
Definition
In the region {s in C: Re(s) > 1},
this infinite series converges and defines a holomorphic function. Bernhard Riemann realized that the zeta function can be extended by analytic continuation in a unique way to a holomorphic function ζ(s) defined for all complex numbers s with s ≠ 1. It is this function that is the object of the Riemann hypothesis.
The connection between this function and prime numbers was already realized by Leonhard Euler:
Relationship to prime numbers
an infinite product extending over all prime numbers p. This is a consequence of the formula for the geometric series and the Fundamental Theorem of Arithmetic.
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.
The zeta function satisfies the following functional equation:
Basic properties
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.
Euler was also able to calculate ζ(2k) for even integers 2k using the formula
One can express the reciprocal of the zeta function using the Möbius function μ(n) as follows:
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 , but we can re-write it as a sum of reciprocals:
Applications
The sum appears to take the form of . However, -1 lies outside of the domain for which Euler's original definition is valid. We must use analytic continuation to define the zeta function at , and when we do, we find that it takes the value .
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: . If, for example, , we can exchange the order of the operators and , 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.
Neal Stephenson's 1999 novel Cryptonomicon mentions as a pseudo-random number source, a useful component in cipher design.
Zeta Functions in Fiction