Riemann zeta function Guide, Meaning , Facts, Information and Description

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

1 Definition 2 Relationship to prime numbers 2.1 An informal justification of the Euler product formula 2.2 The importance of the zeros of ζ(s) 3 Basic properties 4 Applications 5 Zeta Functions in Fiction

Definition

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

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.

Relationship to prime numbers

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

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.

An informal justification of the Euler product formula

A simple way to justify this is to work backwards from Euler's formula to the zeta function. Note: this is a very informal treatment, that omits many technical details. By the normal geometric series formula, each term (for a given prime p) in the product above can be expanded to an infinite sum of terms consisting of the reciprocal of p raised to multiples of s, as follows

Now consider the product of all of the infinite sums above over all the prime numbers. Multiplying through the sums, and then gathering the product terms, we get sums of terms of the form

where p_n is the n'th prime number, and the k_i are integer powers, depending on which terms are gathered from each additive series. We assert, with some vigorous handwaving, that there are terms corresponding to all possible values of positive integer values of k_i, including zero, and each possible term occurs once only.

But by first factoring out s from these terms to put them in the form

and then observing from the fundamental theorem of arithmetic that each of the positive non-zero integers can be uniquely expressed in the form

we end up with terms of the form

for all possible positive non-zero integer n, and we have the original formulation of the zeta function as the sum

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:

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

where B_2k are the Bernoulli numbers. From this one sees that ζ(2); = π²/6, ζ(4) = π⁴/90, ζ(6) = π⁶/945 etc. (sequence A046988/A002432 in OEIS). These give well-known infinite series for π. For odd integers the case is not so simple. Ramanujan did some great work about this.

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

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.

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 , but we can re-write it as a sum of reciprocals:

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 . (There is a famous story regarding Ramanujan and this sum; he had written down the infinite sum in question, and stated that it equalled minus one-twelveth. At first sight, mathematicians looking at it were convinced that the self-taught Ramanujan was simply writing nonsense; it was only when Ramanujan sent some of his results to G. H. Hardy that Hardy and his colleague J. E. Littlewood realised that Ramanujan had reached a deep result about the zeta function without any formal training in mathematics).

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.

Zeta Functions in Fiction

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

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