Riemann zeta function


The Riemann zeta function or Euler–Riemann zeta function,, is a function of a complex variable s that analytically continues the sum of the Dirichlet series
which converges when the real part of is greater than 1. More general [|representations] of for all are given below. The Riemann zeta function plays a pivotal role in analytic number theory and has applications in physics, probability theory, and applied statistics.
As a function of a real variable, Leonhard Euler first introduced and studied it in the first half of the eighteenth century without using complex analysis, which was not available at the time. Bernhard Riemann's 1859 article "On the Number of Primes Less Than a Given Magnitude" extended the Euler definition to a complex variable, proved its meromorphic continuation and functional equation, and established a relation between its zeros and the distribution of prime numbers.
The values of the Riemann zeta function at even positive integers were computed by Euler. The first of them,, provides a solution to the Basel problem. In 1979 Roger Apéry proved the irrationality of. The values at negative integer points, also found by Euler, are rational numbers and play an important role in the theory of modular forms. Many generalizations of the Riemann zeta function, such as Dirichlet series, Dirichlet -functions and -functions, are known.

Definition

The Riemann zeta function is a function of a complex variable.
For the special case where, the zeta function can be expressed by the following integral:
where
is the gamma function.
In the case, the integral for always converges, and can be simplified to the following infinite series:
The Riemann zeta function is defined as the analytic continuation of the function defined for by the sum of the preceding series.
Leonhard Euler considered the above series in 1740 for positive integer values of, and later Chebyshev extended the definition to.
The above series is a prototypical Dirichlet series that converges absolutely to an analytic function for such that and diverges for all other values of. Riemann showed that the function defined by the series on the half-plane of convergence can be continued analytically to all complex values. For the series is the harmonic series which diverges to, and
Thus the Riemann zeta function is a meromorphic function on the whole complex -plane, which is holomorphic everywhere except for a simple pole at with residue 1.

Specific values

For any positive even integer :
where is the th Bernoulli number.
For odd positive integers, no such simple expression is known, although these values are thought to be related to the algebraic -theory of the integers; see Special values of -functions.
For nonpositive integers, one has
for .
In particular, vanishes at the negative even integers because for all odd other than 1. These are the so-called "trivial zeros" of the zeta function.
Via analytic continuation, one can show that:
Taking the limit, one obtains.

Euler product formula

The connection between the zeta function and prime numbers was discovered by Euler, who proved the identity
where, by definition, the left hand side is and the infinite product on the right hand side extends over all prime numbers :
Both sides of the Euler product formula converge for. 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.
The Euler product formula can be used to calculate the asymptotic probability that randomly selected integers are set-wise coprime. Intuitively, the probability that any single number is divisible by a prime is. Hence the probability that numbers are all divisible by this prime is, and the probability that at least one of them is not is. Now, for distinct primes, these divisibility events are mutually independent because the candidate divisors are coprime. Thus the asymptotic probability that numbers are coprime is given by a product over all primes,

Riemann's functional equation

The zeta function satisfies the functional equation:
where is the gamma function. This is an equality of meromorphic functions valid on the whole complex plane. The equation relates values of the Riemann zeta function at the points and, in particular relating even positive integers with odd negative integers. Owing to the zeros of the sine function, the functional equation implies that has a simple zero at each even negative integer, known as the trivial zeros of. When is an even positive integer, the product on the right is non-zero because has a simple pole, which cancels the simple zero of the sine factor.
A proof of the functional equation proceeds as follows:
We observe that if, then
As a result, if then
With the inversion of the limiting processes justified by absolute convergence
For convenience, let
Then
Given that
Then
Hence
This is equivalent to
Or
which is convergent for all s, so holds by analytic continuation. Furthermore, the RHS is unchanged if s is changed to 1 − s. Hence
which is the functional equation.
Attributed to Bernhard Riemann.
The 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 had been conjectured by Euler over a hundred years earlier, in 1749, for the Dirichlet eta function :
Incidentally, this relation gives an equation for calculating in the region 0 < < 1, i.e.
where the η-series is convergent in the larger half-plane .
Riemann also found a symmetric version of the functional equation applying to the xi-function:
which satisfies:

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 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 one of the greatest unsolved problems in mathematics, asserts that any non-trivial zero has. 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 -function.

The Hardy–Littlewood conjectures

In 1914, Godfrey Harold Hardy proved that has infinitely many real zeros.
Hardy and John Edensor Littlewood formulated two conjectures on the density and distance between the zeros of on intervals of large positive real numbers. In the following, is the total number of real zeros and the total number of zeros of odd order of the function lying in the interval.
These two conjectures opened up new directions in the investigation of the Riemann zeta function.

Zero-free region

The location of the Riemann zeta function's zeros is of great importance in the theory of numbers. The prime number theorem is equivalent to the fact that there are no zeros of the zeta function on the line. A better result that follows from an effective form of Vinogradov's mean-value theorem is that whenever 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.

Other results

It is known that there are infinitely many zeros on the critical line. Littlewood showed that if the sequence contains the imaginary parts of all zeros in the upper half-plane in ascending order, then
The critical line theorem asserts that a positive proportion of the nontrivial zeros lies on the critical line.
In the critical strip, the zero with smallest non-negative imaginary part is . The fact that
for all complex implies that the zeros of the Riemann zeta function are symmetric about the real axis. Combining this symmetry with the functional equation, furthermore, one sees that the non-trivial zeros are symmetric about the critical line.

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 :
for every complex number with real part greater than 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 is greater than.

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. The first proof of universality was provided by Sergei Mikhailovitch Voronin in 1975. More recent work has included effective versions of Voronin's theorem and extending it to Dirichlet L-functions.

Estimates of the maximum of the modulus of the zeta function

Let the functions and be defined by the equalities
Here is a sufficiently large positive number,,,,. Estimating the values and from below shows, how large values can take on short intervals of the critical line or in small neighborhoods of points lying in the critical strip.
The case was studied by Kanakanahalli Ramachandra; the case, where is a sufficiently large constant, is trivial.
Anatolii Karatsuba proved, in particular, that if the values and exceed certain sufficiently small constants, then the estimates
hold, where and are certain absolute constants.

The argument of the Riemann zeta function

The function
is called the argument of the Riemann zeta function. Here is the increment of an arbitrary continuous branch of along the broken line joining the points, and.
There are some theorems on properties of the function. Among those results are the mean value theorems for and its first integral
on intervals of the real line, and also the theorem claiming that every interval for
contains at least
points where the function changes sign. Earlier similar results were obtained by Atle Selberg for the case

Representations

Dirichlet series

An extension of the area of convergence can be obtained by rearranging the original series. The series
converges for, while
converges even for. In this way, the area of convergence can be extended to for any negative integer.

Mellin-type integrals

The Mellin transform of a function is defined as
in the region where the integral is defined. There are various expressions for the zeta-function as Mellin transform-like integrals. If the real part of is greater than one, we have
where denotes the gamma function. By modifying the contour, Riemann showed that
for all .
Starting with the integral formula one can show by substitution and iterated differentation for natural
using the notation of umbral calculus where each power is to be replaced by, so e.g. for we have while for this becomes
We can also find expressions which relate to prime numbers and the prime number theorem. If is the prime-counting function, then
for values with.
A similar Mellin transform involves the Riemann prime-counting function, which counts prime powers with a weight of, 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 can be recovered from it by Möbius inversion.

Theta functions

The Riemann zeta function can be given by a Mellin transform
in terms of Jacobi's theta function
However, this integral only converges if the real part of is greater than 1, but it can be regularized. This gives the following expression for the zeta function, which is well defined for all except 0 and 1:

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 is then
The constants here are called the Stieltjes constants and can be defined by the limit
The constant term is the Euler–Mascheroni constant.

Integral

For all,, the integral relation
holds true, which may be used for a numerical evaluation of the zeta-function.

Rising factorial

Another series development using the rising factorial valid for the entire complex plane is
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 ; 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, the trivial zeros at −2, −4, ... due to the gamma function term in the denominator, and the non-trivial zeros at.

Globally convergent series

A globally convergent series for the zeta function, valid for all complex numbers except for some integer, was conjectured by Konrad Knopp and proven 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 discussed by Jonathan Sondow in 1994.
Hasse also proved the globally converging series
in the same publication. Research by Iaroslav Blagouchine
has found that a similar, equivalent series was published by Joseph Ser in 1926. Other similar globally convergent series include
where are the harmonic numbers, are the Stirling numbers of the first kind, is the Pochhammer symbol, are the Gregory coefficients, are the Gregory coefficients of higher order, are the Cauchy numbers of the second kind, and
are the Bernoulli polynomials of the second kind, see Blagouchine's paper.
Peter Borwein has developed an algorithm that applies Chebyshev polynomials to the Dirichlet eta function to produce a very rapidly convergent series suitable for high precision numerical calculations.

Series representation at positive integers via the primorial

Here is the primorial sequence and is Jordan's totient function.

Series representation by the incomplete poly-Bernoulli numbers

The function can be represented, for, by the infinite series
where, is the th branch of the Lambert -function, and is an incomplete poly-Bernoulli number.

The Mellin transform of the Engel map

The function : is iterated to find the coefficients appearing in Engel expansions.
The Mellin transform of the map is related to the Riemann zeta function by the formula

Numerical algorithms

For , the Riemann zeta function has for fixed and for all the following representation in terms of three absolutely and uniformly converging series,where for positive integer one has to take the limit value . The derivatives of can be calculated by differentiating the above series termwise. From this follows an algorithm which allows to compute, to arbitrary precision, and its derivatives using at most summands for any, with explicit error bounds. For, these are as follows:
For a given argument with and one can approximate to any accuracy by summing the first series to, to and neglecting, if one chooses as the next higher integer of the unique solution of in the unknown, and from this. For one can neglect altogether. Under the mild condition one needs at most summands. Hence this algorithm is essentially as fast as the Riemann-Siegel formula. Similar algorithms are possible for Dirichlet L-functions.
In February 2020, Sandeep Tyagi showed that a quantum computer can evaluate in the critical strip with computational complexity that is polylogarithmic in. Following work by Ghaith Ayesh Hiary, the required exponential sums may be rescaled as, for integer.

Applications

The zeta function occurs in applied statistics.
Zeta function regularization is used as one possible means of regularization of divergent series and divergent integrals in quantum field theory. In one notable example, the Riemann
zeta-function shows up explicitly in one method of calculating the Casimir effect. The zeta function is also useful for the analysis of dynamical systems.

Infinite series

The zeta function evaluated at equidistant positive integers appears in infinite series representations of a number of constants.
In fact the even and odd terms give the two sums
and
Parametrized versions of the above sums are given by
and
with and where and are the polygamma function and Euler's constant, as well as
all of which are continuous at. Other sums include
where denotes the imaginary part of a complex number.
There are yet more formulas in the article Harmonic number.

Generalizations

There are a number of related zeta functions that can be considered to be generalizations of the Riemann zeta function. These include the Hurwitz zeta function
, which coincides with the Riemann zeta function when , the Dirichlet -functions and the Dedekind zeta-function. For other related functions see the articles zeta function and -function.
The polylogarithm is given by
which coincides with the Riemann zeta function when.
The Lerch transcendent is given by
which coincides with the Riemann zeta function when and .
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 -dimensional complex space. The special values taken by these functions at positive integer arguments are called multiple zeta values by number theorists and have been connected to many different branches in mathematics and physics.