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Riemann zeta function
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In mathematics, the Riemann zeta function, named after German mathematician Bernhard Riemann, is a prominent function of great significance 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.
The Riemann hypothesis, a conjecture about the distribution of the zeros of the Riemann zeta function, is considered by many mathematicians to be the most important unsolved problem in pure mathematics.
Riemann zeta-function is the function of a complex variable initially defined by the following infinite series:
As a Dirichlet series with bounded coefficient sequence this series converges absolutely to an analytic function on the open half-plane of s such that and diverges on the open half-plane of s such that .
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Encyclopedia
In mathematics, the Riemann zeta function, named after German mathematician Bernhard Riemann, is a prominent function of great significance 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.
The Riemann hypothesis, a conjecture about the distribution of the zeros of the Riemann zeta function, is considered by many mathematicians to be the most important unsolved problem in pure mathematics.
Definition
The Riemann zeta-function is the function of a complex variable initially defined by the following infinite series:
As a Dirichlet series with bounded coefficient sequence this series converges absolutely to an analytic function on the open half-plane of s such that and diverges on the open half-plane of s such that . The function defined by the series on the half-plane of convergence can however be continued analytically to all complex s ? 1. For the series is formally identical to the harmonic series which diverges to infinity. As a result, the zeta function becomes a meromorphic function of the complex variable s, which is holomorphic in the region } of the complex plane and has a simple pole at with residue 1.
Specific values
The values of the zeta function obtained from integral arguments are called zeta constants. The following are the most commonly used values of the Riemann zeta function.
this is the harmonic series.
this is employed in calculating the critical temperature for a Bose–Einstein condensate in physics, and for spin-wave physics in magnetic systems.
the demonstration of this equality is known as the Basel problem. The reciprocal of this sum answers the question: What is the probability that two numbers selected at random are relatively prime?
this is called Apéry's constant.
Stefan–Boltzmann law and Wien approximation in physics.
For any positive even number z, the representation of can be obtained using the following procedure. First, we define by
on U − , where U is a sufficiently small open disk around 0. Then, using well-known property about residue, we obtain
The meaning of the above formula is that right hand side of the equation converges and the value is equal to that of left hand side.
Next, it is known that function has the following property:
- For arbitrary integer m = 1,
Therefore, the equation
holds. This implies that over U − , for any k = 0, the kth differential is a rational function of rational numbers, , and . The differentials for k = 1, 2 and 3 are shown below as an example:
For arbitrary k, each trigonometric functions in these k-th differential are expanded to Taylor series which absolutely converges on U. Applying calculus of power series, a rational formula containing only rational numbers and in each term, and which is equal to
is obtained. Therefore, itself is equal to rational formula containing rational numbers and .
Euler product formula
The connection between the zeta function and prime numbers was discovered by Leonhard Euler, who proved the identity
where, by definition, the left hand side is ?(s) and the infinite product on the right hand side extends over all prime numbers p (such expressions are called Euler products):
Both sides of the Euler product formula converge for Re(s) > 1. The proof of Euler's identity uses only the formula for the geometric series and the fundamental theorem of arithmetic. Since the harmonic series, obtained when , diverges, Euler's formula implies that there are infinitely many primes.
For s an integer number, the Euler product formula can be used to calculate the probability that s randomly selected integers are relatively prime. It turns out that this probability is indeed 1/?(s).
The functional equation
The Riemann zeta function satisfies the functional equation
valid for all complex numbers s, which relates its values at points s and . Here, G denotes the gamma function. This functional equation was established by Riemann in his 1859 paper On the Number of Primes Less Than a Given Magnitude and used to construct the analytic continuation in the first place. An equivalent relationship was conjectured by Euler in 1749 for the function
-
According to André Weil, Riemann seems to have been very familiar with Euler's work on the subject.
The functional equation given by Riemann has to be interpreted analytically if any factors in the equation have a zero or pole. For instance, when s is 2, the right side has a simple zero in the sine factor and a simple pole in the Gamma factor, which cancel out and leave a nonzero finite value. Similarly, when s is 0, the right side has a simple zero in the sine factor and a simple pole in the zeta factor, which cancel out and leave a finite nonzero value. When s is 1, the right side has a simple pole in the Gamma factor that is not cancelled out by a zero in any other factor, which is consistent with the zeta-function on the left having a simple pole at 1.
There is also a symmetric version of the functional equation, given by first defining
The functional equation is then given by
(Riemann defined a similar but different function which he called ?(t).)
The functional equation also gives the asymptotic limit
(Gergo Nemes, 2007)
Zeros, the critical line, and the Riemann hypothesis
The functional equation shows that the Riemann zeta function has zeros at .. . These are called the trivial zeros. They are trivial in the sense that their existence is relatively easy to prove, for example, from sin(ps/2) being 0 in the functional equation. The non-trivial zeros have captured far more attention because their distribution not only is far less understood but, more importantly, their study yields impressive results concerning prime numbers and related objects in number theory. It is known that any non-trivial zero lies in the open strip , which is called the critical strip. The Riemann hypothesis, considered to be one of the greatest unsolved problems in mathematics, asserts that any non-trivial zero s has Re(s) = 1/2. In the theory of the Riemann zeta function, the set is called the critical line. For the Riemann zeta function on the critical line, see Z-function.
The location of the Riemann zeta function's zeros is of great importance in the theory of numbers. From the fact that all non-trivial zeros lie in the critical strip one can deduce the prime number theorem. A better result is that ? 0 whenever|t| = 3 and
The strongest result of this kind one can hope for is the truth of the Riemann hypothesis, which would have many profound consequences in the theory of numbers.
It is known that there are infinitely many zeros on the critical line. Littlewood showed that if the sequence (?n) contains the imaginary parts of all zeros in the upper half-plane in ascending order, then
The critical line theorem asserts that a positive percentage of the nontrivial zeros lies on the critical line.
In the critical strip, the zero with smallest non-negative imaginary part is .. Directly from the functional equation one sees that the non-trivial zeros are symmetric about the axis . Furthermore, the fact that for all complex (* indicating complex conjugation) implies that the zeros of the Riemann zeta function are symmetric about the real axis.
The statistics of the Riemann zeta zeros are a topic of interest to mathematicians because of their connection to big problems like the Riemann hypothesis, distribution of prime numbers, etc. Through connections with random matrix theory and quantum chaos, the appeal is even broader. The fractal structure of the Riemann zeta zero distribution has been studied using rescaled range analysis. The self-similarity of the zero distribution is quite remarkable, and is characterized by a large fractal dimension of 1.9. This rather large fractal dimension is found over zeros covering at least fifteen orders of magnitude, and also for the zeros of other L-functions.
Concerning applications: the properties of the Riemann zeta function in the complex plane, specifically along parallels to the imaginary axis, have also been studied, by the relation to prime numbers, in recent physical interference experiments, by decomposing the defining sum into two parts with opposite phases, ψ and ψ*, which then are brought to interference.
Various properties
For sums involving the zeta-function at integer and half-integer values, see rational zeta series.
Reciprocal
The reciprocal of the zeta function may be expressed as a Dirichlet series over the Möbius function µ(n):
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 Riemann hypothesis is equivalent to the claim that this expression is valid when the real part of s is greater than 1/2.
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.
Representations
Mellin transform
The Mellin transform of a function f(x) is defined as
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
where G denotes the Gamma function. By subtracting off the first terms of the power series expansion of around zero, we can get the zeta-function in other regions. In particular, in the critical strip we have
and when the real part of s is between −1 and 0,
We can also find expressions which relate to prime numbers and the prime number theorem. If p(x) is the prime-counting function, then
for values with We can relate this to the Mellin transform of p(x) by
-
where
converges for
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
Now we have
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 p(x) can be recovered from it by Möbius inversion.
Also, from the above (specifically, the second equation in this section), we can write the zeta function in the commonly seen form:
Laurent series
The Riemann zeta function is meromorphic with a single pole of order one at
. It can therefore be expanded as a Laurent series about
the series development then is
The constants ?n here are called the Stieltjes constants and can be defined
by the limit
-
The constant term ?0 is the Euler-Mascheroni constant.
Rising factorial
Another series development valid for the entire complex plane is
where is the rising factorial
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 xs−1; that context gives rise to a series expansion in terms of the falling factorial.
Hadamard product
On the basis of Weierstrass's factorization theorem, Hadamard gave the infinite product expansion
where the product is over the non-trivial zeros ? of ? and the letter ? again denotes the Euler-Mascheroni constant. A simpler infinite product expansion is
This form clearly displays the simple pole at s = 1, the trivial zeros at -2, -4, ... due to the gamma function term in the denominator, and the non-trivial zeros at s = ?.
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:
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.
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 + 4 + · · ·, but we can rewrite it as a sum of reciprocals:
-
The sum S appears to take the form of 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 represented 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
where the notation indicates Ramanujan summation.
For even powers we have:
and for odd powers we have a relation with the Bernoulli numbers:
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. These include the Hurwitz zeta function
which coincides with Riemann's zeta-function when q = 1 (note that the lower limit of summation in the Hurwitz zeta function is 0, not 1), the Dirichlet L-functions and the Dedekind zeta-function. For other related functions see the articles Zeta function and L-function.
The polylogarithm is given by
which coincides with Riemann's zeta-function when z = 1.
The Lerch transcendent is given by
which coincides with Riemann's zeta-function when z = 1 and q = 0 (note that the lower limit of summation in the Lerch transcendent is 0, not 1).
The Clausen function that can be chosen as the real or imaginary part of
The multiple zeta functions are defined by
One can analytically continue these functions to the n-dimensional complex space. The special values of these functions are called multiple zeta values by number theorists and have been connected to many different branches in mathematics and physics.
See also
External links
- — an explanation with a more mathematical approach
-
- (To download compressed archive, click on Download Now... button.)
- A general, non-technical description of the significance of the zeta function in relation to prime numbers.
- Visually-oriented investigation of where zeta is real or purely imaginary.
- functions.wolfram.com
- , section 23.2 of Abramowitz and Stegun
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