Riemann zeta function

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Template:Short description

The Riemann zeta function Template:Math plotted with domain coloring<ref>Template:Cite web</ref>
The pole at Template:Math and two zeros on the critical line

The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter Template:Math (zeta), is a mathematical function of a complex variable defined as <math display="block"> \zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s} = \frac{1}{1^s} + \frac{1}{2^s} + \frac{1}{3^s} + \cdots</math> for Template:Math, and its analytic continuation elsewhere.<ref name=":0" />

The Riemann zeta function plays a pivotal role in analytic number theory and has applications in physics, probability theory, and applied statistics.

Leonhard Euler first introduced and studied the function over the reals in the first half of the eighteenth century. 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. This paper also contained the Riemann hypothesis, a conjecture about the distribution of complex zeros of the Riemann zeta function that many mathematicians consider the most important unsolved problem in pure mathematics.<ref>Template:Cite web</ref>

The values of the Riemann zeta function at even positive integers were computed by Euler. The first of them, Template:Math, provides a solution to the Basel problem. In 1979 Roger Apéry proved the irrationality of [[Apéry's constant|Template:Math]]. 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 L-function|Dirichlet Template:Math-functions]] and [[L-function|Template:Math-functions]], are known.

Definition

File:Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse.pdf

The Riemann zeta function Template:Math is a function of a complex variable Template:Math, where Template:Math and Template:Math are real numbers. (The notation Template:Math, Template:Math, and Template:Math is used traditionally in the study of the zeta function, following Riemann.) When Template:Math, the function can be written as a converging summation or as an integral:

<math>\zeta(s) =\sum_{n=1}^\infty\frac{1}{n^s} = \frac{1}{\Gamma(s)} \int_0^\infty \frac{x ^ {s-1}}{e ^ x - 1} \, \mathrm{d}x\,,</math>

where

<math>\Gamma(s) = \int_0^\infty x^{s-1}\,e^{-x} \, \mathrm{d}x </math>

is the gamma function. The Riemann zeta function is defined for other complex values via analytic continuation of the function defined for Template:Math.

Leonhard Euler considered the above series in 1740 for positive integer values of Template:Math, and later Chebyshev extended the definition to Template:Math.<ref name='devlin'>Template:Cite book</ref>

The above series is a prototypical Dirichlet series that converges absolutely to an analytic function for Template:Math such that Template:Math and diverges for all other values of Template:Math. Riemann showed that the function defined by the series on the half-plane of convergence can be continued analytically to all complex values Template:Math. For Template:Math, the series is the harmonic series which diverges to Template:Math, and <math display="block"> \lim_{s \to 1} (s - 1)\zeta(s) = 1.</math> Thus the Riemann zeta function is a meromorphic function on the whole complex plane, which is holomorphic everywhere except for a simple pole at Template:Math with residue Template:Math.

Euler's product formula

In 1737, the connection between the zeta function and prime numbers was discovered by Euler, who proved the identity

<math>\sum_{n=1}^\infty\frac{1}{n^s} = \prod_{p \text{ prime}} \frac{1}{1-p^{-s}},</math>

where, by definition, the left hand side is Template:Math and the infinite product on the right hand side extends over all prime numbers Template:Math (such expressions are called Euler products):

<math>\prod_{p \text{ prime}} \frac{1}{1-p^{-s}} = \frac{1}{1-2^{-s}}\cdot\frac{1}{1-3^{-s}}\cdot\frac{1}{1-5^{-s}}\cdot\frac{1}{1-7^{-s}}\cdot\frac{1}{1-11^{-s}} \cdots \frac{1}{1-p^{-s}} \cdots</math>

Both sides of the Euler product formula converge for Template:Math. 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 Template:Math, diverges, Euler's formula (which becomes Template:Math) implies that there are infinitely many primes.<ref>Template:Cite book</ref> Since the logarithm of Template:Math is approximately Template:Math, the formula can also be used to prove the stronger result that the sum of the reciprocals of the primes is infinite. On the other hand, combining that with the sieve of Eratosthenes shows that the density of the set of primes within the set of positive integers is zero.

The Euler product formula can be used to calculate the asymptotic probability that Template:Math randomly selected integers within a bound are set-wise coprime. Intuitively, the probability that any single number is divisible by a prime (or any integer) Template:Math is Template:Math. Hence the probability that Template:Math numbers are all divisible by this prime is Template:Math, and the probability that at least one of them is not is Template:Math. Now, for distinct primes, these divisibility events are mutually independent because the candidate divisors are coprime (a number is divisible by coprime divisors Template:Math and Template:Math if and only if it is divisible by Template:Math, an event which occurs with probability Template:Math). Thus the asymptotic probability that Template:Math numbers are coprime is given by a product over all primes,<ref>Template:Cite book</ref>

<math>\prod_{p \text{ prime}} \left(1-\frac{1}{p^s}\right) = \left( \prod_{p \text{ prime}} \frac{1}{1-p^{-s}} \right)^{-1} = \frac{1}{\zeta(s)}. </math>

Riemann's functional equation

This zeta function satisfies the functional equation <math display="block"> \zeta(s) = 2^s \pi^{s-1}\ \sin\left( \frac{\pi s}{2} \right)\ \Gamma(1-s)\ \zeta(1-s)\ ,</math> where Template:Math 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 Template:Math and Template:Math, in particular relating even positive integers with odd negative integers. Owing to the zeros of the sine function, the functional equation implies that Template:Math has a simple zero at each even negative integer Template:Math, known as the trivial zeros of Template:Math. When Template:Math is an even positive integer, the product Template:Nobr on the right is non-zero because Template:Math has a simple pole, which cancels the simple zero of the sine factor. When Template:Math is Template:Math, the zero of the sine factor is cancelled by the simple pole of Template:Math.

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A proof of the functional equation proceeds as follows: We observe that if Template:Math, then <math display="block"> \int_0^\infty x^{ \frac{1}{2} s - 1 } e^{-n^2\pi x}\ \mathrm dx\ =\ \frac{\ \Gamma\!\left( \frac{s}{2} \right)\ }{\ n^s\ \pi^{\frac{s}{2}}\ } ~.</math>

As a result, if Template:Math then <math display="block"> \frac{\ \Gamma\!\left(\frac{s}{2}\right)\ \zeta(s)\ }{\ \pi^{ \frac{s}{2} }\ }\ =\ \sum_{n=1}^\infty\ \int_0^\infty\ x^{{s\over 2}-1}\ e^{-n^2 \pi x}\ \mathrm dx\ =\ \int_0^\infty x^{{s\over 2}-1} \sum_{n=1}^\infty e^{-n^2 \pi x}\ \mathrm dx\ ,</math> with the inversion of the limiting processes justified by absolute convergence (hence the stricter requirement on <math>s</math>).

For convenience, let <math display="block"> \psi(x)\ := \ \sum_{n=1}^\infty\ e^{-n^2 \pi x} ,</math> which is a special case of the theta function.

Because <math>e^{-n^2 \pi x}</math> and <math>\frac1\sqrt{x} e^{\frac{-n^2 \pi}{x}}</math> are Fourier transform pairs,<ref name='Damm-Johnsen'>Template:Cite book</ref> then, by the Poisson summation formula, we have <math display="block"> \sum_{n=-\infty}^\infty\ e^{ - n^2 \pi\ x }\ =\ \frac{ 1 }{\ \sqrt{x\ }\ }\ \sum_{n=-\infty}^\infty\ e^{ -\frac{\ n^2 \pi\ }{ x } }\ ,</math> so that <math display="block">\ 2\ \psi(x) + 1\ =\ \frac{ 1 }{\ \sqrt{x\ }\ } \left(\ 2\ \psi\!\left( \frac{ 1 }{ x } \right) + 1\ \right) ~.</math>

Hence <math display="block"> \pi^{ -\frac{s}{2} }\ \Gamma\!\left( \frac{s}{2} \right)\ \zeta(s)\ =\ \int_0^1\ x^{ \frac{s}{2} - 1 }\ \psi(x)\ \mathrm dx + \int_1^\infty x^{ \frac{s}{2} - 1 } \psi(x)\ \mathrm dx ~.</math>

The right side is equivalent to <math display="block"> \int_0^1 x^{ \frac{s}{2} - 1 } \left( \frac{ 1 }{\ \sqrt{x\ }\ }\ \psi\!\left( \frac{1}{x} \right) + \frac{ 1 }{\ 2 \sqrt{x\ }\ } - \frac{ 1 }{ 2 }\ \right) \ \mathrm dx + \int_1^\infty x^{{s\over 2}-1} \psi(x)\ \mathrm dx </math> or <math display="block"> \frac{ 1 }{\ s - 1\ } - \frac{ 1 }{\ s\ } + \int_0^1\ x^{ \frac{s}{2} - \frac{3}{2}}\ \psi\!\left( \frac{ 1 }{\ x\ } \right)\ \mathrm dx + \int_1^\infty\ x^{ \frac{s}{2} - 1 }\ \psi(x)\ \mathrm dx ~.</math>

So <math display="block"> \pi^{ -\frac{ s }{ 2 } }\ \Gamma\!\left( \frac{\ s\ }{ 2 } \right)\ \zeta(s)\ =\ \frac{ 1 }{\ s ( s - 1 )\ } + \int_1^\infty\ \left( x^{ -\frac{ s }{ 2 } - \frac{ 1 }{ 2 } } + x^{ \frac{ s }{ 2 } - 1 } \right)\ \psi(x)\ \mathrm dx </math> which is convergent for all Template:Math, because Template:Math more quickly than any power of Template:Math for Template:Math, so the integral converges. As the RHS remains the same if Template:Math is replaced by Template:Math, <math display="block"> \frac{\ \Gamma\!\left(\ \frac{s}{2}\ \right)\ \zeta\!\left(\ s\ \right)\ }{\ \pi^{ \frac{s}{2}\ }\ }\ =\ \frac{\ \Gamma\!\left(\ \frac{1}{2} - \frac{s}{2}\ \right)\ \zeta\!\left(\ 1 - s\ \right)\ }{\ \pi^{ \frac{1}{2} - \frac{s}{2} }\ } </math> which is the functional equation attributed to Bernhard Riemann.<ref>Template:Cite book</ref>

The functional equation above can be obtained using both the reflection formula and the duplication formula.

First collect terms of Template:Math: <math display="block">\Gamma\left(\frac{s}{2}\right)\zeta\left(s\right) = \Gamma\left(\frac{1}{2} - \frac{s}{2}\right)\zeta\left(1 - s\right)\pi^{s-\frac{1}{2}}</math>

Then multiply both sides by Template:Math and use the reflection formula: <math display="block">\Gamma\left(1-\frac s2\right)\Gamma\left(\frac{s}{2}\right)\zeta\left(s\right) = \Gamma\left(1-\frac s2\right)\Gamma\left(\frac{1}{2} - \frac{s}{2}\right)\zeta\left(1 - s\right)\pi^{s-\frac{1}{2}}</math>

<math display="block">\zeta\left(s\right) = \sin\left(\frac{\pi s}2\right)\Gamma\left(1-\frac s2\right)\Gamma\left(\frac{1}{2} - \frac{s}{2}\right)\zeta\left(1 - s\right)\pi^{s-\frac{3}{2}}</math>

Use the duplication formula with Template:Math <math display="block">\zeta\left(s\right) = \sin\left(\frac{\pi s}2\right)2^{1-1+s}\sqrt{\pi}\Gamma\left(1-s\right)\zeta\left(1 - s\right)\pi^{s-\frac{3}{2}}</math> so that <math display="block">\zeta\left(s\right) = \sin\left(\frac{\pi s}2\right)2^s\Gamma\left(1-s\right)\zeta\left(1 - s\right)\pi^{s-1}</math>

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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.

Riemann's xi function

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Riemann also found a symmetric version of the functional equation by setting <math display="block">\xi(s) =\frac{s(s-1)}{2} \times \pi^{-\frac{s}{2}}\Gamma\left( \frac{s}{2} \right)\zeta(s) = (s-1)\pi^{-\frac{s}{2}}\Gamma\left(\frac{s}{2}+1\right)\zeta(s)</math> that satisfies: <math display="block"> \xi(s) = \xi(1 - s) ~.</math>

Returning to the functional equation's derivation in the previous section, we have <math display="block"> \xi(s) =\frac12 + \frac{s(s-1)}{2} \int_1^\infty \left(x^{-\frac{s}{2}-\frac{1}{2}} + x^{\frac{s}{2}-1}\right)\psi(x) dx </math>

Using integration by parts, <math display="block"> \xi(s) =\frac12 - \left[\left(sx^{\frac{1-s}{2}} + (1-s)x^{\frac{s}{2}}\right)\psi(x)\right]_1^\infty + \int_1^\infty \left(sx^{\frac{1-s}{2}} + (1-s)x^{\frac{s}{2}}\right)\psi'(x) dx </math> <math display="block"> \xi(s) =\frac12 + \psi(1) + \int_1^\infty \left(sx^{\frac{1-s}{2}} + (1-s)x^{\frac{s}{2}}\right)\psi'(x) dx </math>

Using integration by parts again with a factorization of Template:Math, <math display="block"> \xi(s) =\frac12 + \psi(1) - 2\left[x^{\frac32}\psi'(x)\left(x^{\frac{s-1}{2}} + x^{-\frac{s}{2}}\right)\right]_1^\infty + 2\int_1^\infty \left(x^{\frac{s-1}{2}} + x^{-\frac{s}{2}}\right)\frac{d}{dx}\left[x^{\frac32}\psi'(x)\right] dx </math> <math display="block"> \xi(s) =\frac12 +\psi(1) + 4\psi'(1) + 2\int_1^\infty \frac{d}{dx}\left[x^{\frac32}\psi'(x)\right]\left(x^{\frac{s-1}{2}} + x^{-\frac{s}{2}}\right) dx </math>

As <math>\frac12 +\psi(1) + 4\psi'(1)=0</math>, <math display="block"> \xi(s) = 2\int_1^\infty \frac{d}{dx}\left[x^{\frac32}\psi'(x)\right]\left(x^{\frac{s-1}{2}} + x^{-\frac{s}{2}}\right) dx </math>

Remove a factor of Template:Math to make the exponents in the remainder opposites. <math display="block"> \xi(s) = 2\int_1^\infty \frac{d}{dx}\left[x^{\frac32}\psi'(x)\right]x^{-\frac14}\left(x^{\frac{s-1/2}{2}} + x^{\frac{1/2-s}{2}}\right) dx </math>

Using the hyperbolic functions, namely Template:Math, and letting Template:Math gives <math display="block"> \xi(s) = 4\int_1^\infty \frac{d}{dx}\left[x^{\frac32}\psi'(x)\right]x^{-\frac14}\cos(\frac{t}2\log x) dx </math> and by separating the integral and using the power series for Template:Math, <math display="block"> \xi(s) = \sum_{n=0}^\infty a_{2n}t^{2n} </math> which led Riemann to his famous hypothesis.

Zeros, the critical line, and the Riemann hypothesis

Template:Main

The Riemann zeta function has no zeros to the right of Template:Math or (apart from the trivial zeros) to the left of Template:Math (nor can the zeros lie too close to those lines). Furthermore, the non-trivial zeros are symmetric about the real axis and the line Template:Math and, according to the Riemann hypothesis, they all lie on the line Template:Math.
This image shows a plot of the Riemann zeta function along the critical line for real values of Template:Math running from 0 to 34. The first five zeros in the critical strip are clearly visible as the place where the spirals pass through the origin.
The real part (red) and imaginary part (blue) of the Riemann zeta function along the critical line Template:Math. The first non-trivial zeros can be seen at Template:Math Template:Math, Template:Math and Template:Math.

The functional equation shows that the Riemann zeta function has zeros at Template:Math. These are called the trivial zeros. They are trivial in the sense that their existence is relatively easy to prove, for example, from Template:Math being Template:Math 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 important results concerning prime numbers and related objects in number theory. It is known that any non-trivial zero lies in the open strip Template:Math, which is called the critical strip. The set Template:Math is called the critical line. The Riemann hypothesis, considered one of the greatest unsolved problems in mathematics, asserts that all non-trivial zeros are on the critical line. In 1989, Conrey proved that more than 40% of the non-trivial zeros of the Riemann zeta function are on the critical line.<ref>Template:Cite journal</ref> This has since been improved to 41.7%.<ref>Template:Cite journal</ref>

For the Riemann zeta function on the critical line, see [[Z function|Template:Math-function]].

First few nontrivial zeros<ref>Template:Cite web</ref><ref>Template:Cite web</ref>
Zero
Template:Math
Template:Math
Template:Math
Template:Math
Template:Math
Template:Math
Template:Math

Number of zeros in the critical strip

Let Template:Math be the number of zeros of Template:Math in the critical strip Template:Math, whose imaginary parts are in the interval Template:Math. Timothy Trudgian proved that, if Template:Math, then<ref>Template:Cite journal</ref>

<math> \left|N(T) - \frac{T}{2\pi} \log{\frac{T}{2\pi e}}\right| \leq 0.112 \log T + 0.278 \log\log T + 3.385 + \frac{0.2}{T}</math>.

Hardy–Littlewood conjectures

In 1914, G. H. Hardy proved that Template:Math has infinitely many real zeros.<ref>Template:Cite journal</ref><ref>Template:Cite journal</ref>

Hardy and J. E. Littlewood formulated two conjectures on the density and distance between the zeros of Template:Math on intervals of large positive real numbers. In the following, Template:Math is the total number of real zeros and Template:Math the total number of zeros of odd order of the function Template:Math lying in the interval Template:Math. Template:Numbered list 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 number theory. The prime number theorem is equivalent to the fact that there are no zeros of the zeta function on the line Template:Math.<ref name="Diamond1982">Template:Cite journal</ref> It is also known that zeros do not exist in certain regions slightly to the left of the line Template:Math, known as zero-free regions. For instance, Korobov<ref>Template:Cite journal</ref> and Vinogradov<ref>Template:Cite journal</ref> independently showed via the Vinogradov's mean-value theorem that for sufficiently large Template:Math, Template:Math for

<math>\sigma \geq 1 - \frac{c}{(\log|t|)^{2/3 + \varepsilon}}</math>

for any Template:Math and a number Template:Math depending on Template:Math. Asymptotically, this is the largest known zero-free region for the zeta function.

Explicit zero-free regions are also known. Platt and Trudgian<ref>Template:Cite journal</ref> verified computationally that Template:Math if Template:Math and Template:Math. Mossinghoff, Trudgian and Yang proved<ref>Template:Cite journal</ref> that zeta has no zeros in the region

<math>\sigma\ge 1 - \frac{1}{5.558691\log|t|}</math>

for Template:Math, which is the largest known zero-free region in the critical strip for Template:Math (for previous results see<ref>Template:Cite journal</ref>). Yang<ref>Template:Cite journal</ref> showed that Template:Math if

<math>\sigma \geq 1 - \frac{\log\log|t|}{21.233\log|t|}</math> and <math>|t| \geq 3</math>

which is the largest known zero-free region for Template:Math. Bellotti proved<ref>Template:Cite journal</ref> (building on the work of Ford<ref>Template:Cite journal</ref>) the zero-free region

<math>\sigma \ge 1 - \frac{1}{53.989(\log|t|)^{2/3}(\log\log|t|)^{1/3}}</math> and <math>|t| \ge 3</math>.

This is the largest known zero-free region for fixed Template:Math. Bellotti also showed that for sufficiently large Template:Math, the following better result is known: Template:Math for

<math>\sigma \geq 1 - \frac{1}{48.0718(\log|t|)^{2/3}(\log\log|t|)^{1/3}}.</math>

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 (Template:Math) contains the imaginary parts of all zeros in the upper half-plane in ascending order, then

<math>\lim_{n\rightarrow\infty}\left(\gamma_{n+1}-\gamma_n\right)=0.</math>

The critical line theorem asserts that a positive proportion of the nontrivial zeros lies on the critical line. (The Riemann hypothesis would imply that this proportion is Template:Math.)

In the critical strip, the zero with smallest non-negative imaginary part is Template:Math (Template:OEIS2C). The fact that, for all complex Template:Math,

<math>\zeta(s)=\overline{\zeta(\overline{s})}</math>

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 Template:Math.

It is also known that no zeros lie on the line with real part Template:Math.

A large class of modified zeta functions exists that share the same non-trivial zeros as the Riemann zeta function, where modification means replacing the prime numbers in the Euler product by real numbers, which was shown in a result by Grosswald and Schnitzer.

Specific values

Template:Main For any positive even integer Template:Math, <math display="block"> \zeta(2n) = \frac{|{B_{2n}}|(2\pi)^{2n}}{2(2n)!},</math> where Template:Math is the Template:Mathth Bernoulli number. For odd positive integers, no such simple expression is known, although these values are thought to be related to the algebraic Template:Math-theory of the integers; see [[Special values of L-functions|Special values of Template:Math-functions]].

For nonpositive integers, one has <math display="block">\zeta(-n)= -\frac{B_{n+1}}{n+1}</math> for Template:Math (using the convention that Template:Math). In particular, Template:Math vanishes at the negative even integers because Template:Math for all odd Template:Math other than Template:Math. These are the so-called "trivial zeros" of the zeta function.

Via analytic continuation, one can show that <math display="block">\zeta(-1) = -\tfrac{1}{12}</math> This gives a pretext for assigning a finite value to the divergent series Template:Math, which has been used in certain contexts (Ramanujan summation) such as string theory.<ref name='polchinski'>Template:Cite book</ref> Analogously, the particular value <math display="block">\zeta(0) = -\tfrac{1}{2}</math> can be viewed as assigning a finite result to the divergent series Template:Math.

The value <math display="block">\zeta\bigl(\tfrac12\bigr) = -1.46035450880958681288\ldots</math> is employed in calculating kinetic boundary layer problems of linear kinetic equations.<ref>Template:Cite journal</ref><ref>Further digits and references for this constant are available at Template:OEIS2C.</ref>

Although <math display="block">\zeta(1) = 1 + \tfrac{1}{2} + \tfrac{1}{3} + \cdots</math> diverges, its Cauchy principal value <math display="block"> \lim_{\varepsilon \to 0} \frac{\zeta(1+\varepsilon)+\zeta(1-\varepsilon)}{2}</math> exists and is equal to the Euler–Mascheroni constant Template:Math.<ref name=Sondow1998>Template:Cite journal</ref>

The demonstration of the particular value <math display="block">\zeta(2) = 1 + \frac{1}{2^2} + \frac{1}{3^2} + \cdots = \frac{\pi^2}{6}</math> is known as the Basel problem. The reciprocal of this sum answers the question: 'What is the probability that two numbers selected from a uniform distribution from Template:Math to Template:Math] are coprime as Template:Math?'<ref>Template:Cite book</ref> The value <math display="block">\zeta(3) = 1 + \frac{1}{2^3} + \frac{1}{3^3} + \cdots = 1.202056903159594285399...</math> is Apéry's constant.

Taking the limit Template:Math through the real numbers, one obtains Template:Math. But at complex infinity on the Riemann sphere the zeta function has an essential singularity.<ref name=":0">Template:Cite journal</ref>

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 Template:Math:

<math>\frac{1}{\zeta(s)} = \sum_{n=1}^\infty \frac{\mu(n)}{n^s}</math>

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

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.<ref>Template:Cite journal Reprinted in Math. USSR Izv. (1975) 9: 443–445.</ref> More recent work has included effective versions of Voronin's theorem<ref>Template:Cite journal</ref> and extending it to [[Dirichlet L-function|Dirichlet Template:Math-function]]s.<ref>Template:Cite journal</ref><ref>Template:Cite book</ref>

Estimates of the maximum of the modulus of the zeta function

Let the functions Template:Math and Template:Math be defined by the equalities

<math> F(T;H) = \max_{|t-T|\le H}\left|\zeta\left(\tfrac{1}{2}+it\right)\right|,\qquad G(s_{0};\Delta) = \max_{|s-s_{0}|\le\Delta}|\zeta(s)|. </math>

Here Template:Math is a sufficiently large positive number, Template:Math, Template:Math, Template:Math, Template:Math. Estimating the values Template:Math and Template:Math from below shows, how large (in modulus) values Template:Math can take on short intervals of the critical line or in small neighborhoods of points lying in the critical strip Template:Math.

The case Template:Math was studied by Kanakanahalli Ramachandra; the case Template:Math, where Template:Math is a sufficiently large constant, is trivial.

Anatolii Karatsuba proved,<ref>Template:Cite journal</ref><ref>Template:Cite journal</ref> in particular, that if the values Template:Math and Template:Math exceed certain sufficiently small constants, then the estimates

<math> F(T;H) \ge T^{- c_1},\qquad G(s_0; \Delta) \ge T^{-c_2}, </math>

hold, where Template:Math and Template:Math are certain absolute constants.

Argument of the Riemann zeta function

The function

<math>S(t) = \frac{1}{\pi}\arg{\zeta\left(\tfrac12+it\right)}</math>

is called the argument of the Riemann zeta function. Here Template:Math is the increment of an arbitrary continuous branch of Template:Math along the broken line joining the points Template:Math, Template:Math and Template:Math.

There are some theorems on properties of the function Template:Math. Among those results<ref>Template:Cite journal</ref><ref>Template:Cite journal</ref> are the mean value theorems for Template:Math and its first integral

<math>S_1(t) = \int_0^t S(u) \, \mathrm{d}u</math>

on intervals of the real line, and also the theorem claiming that every interval Template:Math for

<math>H \ge T^{\frac{27}{82}+\varepsilon}</math>

contains at least

<math> H\sqrt[3]{\ln T}e^{-c\sqrt{\ln\ln T}} </math>

points where the function Template:Math changes sign. Earlier similar results were obtained by Atle Selberg for the case

<math>H\ge T^{\frac12+\varepsilon}.</math>

Representations

Dirichlet series

An extension of the area of convergence can be obtained by rearranging the original series.<ref name="Knopp">Template:Cite book</ref> The series

<math>\zeta(s)=\frac{1}{s-1}\sum_{n=1}^\infty \left(\frac{n}{(n+1)^s}-\frac{n-s}{n^s}\right)</math>

converges for Template:Math, while

<math>\zeta(s) =\frac{1}{s-1}\sum_{n=1}^\infty\frac{n(n+1)}{2}\left(\frac{2n+3+s}{(n+1)^{s+2}}-\frac{2n-1-s}{n^{s+2}}\right)</math>

converge even for Template:Math. In this way, the area of convergence can be extended to Template:Math for any negative integer Template:Math.

The recurrence connection is clearly visible from the expression valid for Template:Math enabling further expansion by integration by parts.

<math>\begin{aligned}

\zeta(s)= & 1+\frac{1}{s-1}-\frac{s}{2 !}[\zeta(s+1)-1] \\ - & \frac{s(s+1)}{3 !}[\zeta(s+2)-1] \\ & -\frac{s(s+1)(s+2)}{3 !} \sum_{n=1}^{\infty} \int_0^1 \frac{t^3 d t}{(n+t)^{s+3}}. \end{aligned}</math> This recurrence leads to this other series development that uses the rising factorial and is valid for the entire complex plane <ref name="Knopp" />

<math>\zeta(s) = \frac{s}{s-1} - \sum_{n=1}^\infty \bigl(\zeta(s+n)-1\bigr)\frac{s(s+1)\cdots(s+n-1)}{(n+1)!}.</math>

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 Template:Math; that context gives rise to a series expansion in terms of the falling factorial.<ref>Template:Cite web</ref>

Mellin-type integrals

The Mellin transform of a function Template:Math is defined as<ref>Template:Cite journal translated and reprinted in Template:Cite book</ref>

<math> \int_0^\infty f(x)x^s\, \frac{\mathrm{d}x}{x} </math>

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 Template:Math is greater than one, we have

<math>\Gamma(s)\zeta(s) =\int_0^\infty\frac{x^{s-1}}{e^x-1} \,\mathrm{d}x \quad</math> and <math>\quad\Gamma(s)\zeta(s) =\frac1{2s}\int_0^\infty\frac{x^{s}}{\cosh(x)-1} \,\mathrm{d}x ,</math>

where Template:Math denotes the gamma function. By modifying the contour, Riemann showed that

<math>2\sin(\pi s)\Gamma(s)\zeta(s) =i\oint_H \frac{(-x)^{s-1}}{e^x-1}\,\mathrm{d}x </math>

for all Template:Math<ref>Trivial exceptions of values of Template:Math that cause removable singularities are not taken into account throughout this article.</ref> (where Template:Math denotes the Hankel contour).

We can also find expressions which relate to prime numbers and the prime number theorem. If Template:Math is the prime-counting function, then

<math>\ln \zeta(s) = s \int_0^\infty \frac{\pi(x)}{x(x^s-1)}\,\mathrm{d}x,</math>

for values with Template:Math.

A similar Mellin transform involves the Riemann function Template:Math, which counts prime powers Template:Math with a weight of Template:Math, so that

<math>J(x) = \sum \frac{\pi\left(x^\frac{1}{n}\right)}{n}.</math>

Now

<math>\ln \zeta(s) = s\int_0^\infty J(x)x^{-s-1}\,\mathrm{d}x. </math>

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 Template:Math can be recovered from it by Möbius inversion.

Theta functions

The Riemann zeta function can be given by a Mellin transform<ref>Template:Cite book</ref>

<math>2\pi^{-\frac{s}{2}}\Gamma\left(\frac{s}{2}\right)\zeta(s) = \int_0^\infty \bigl(\theta(it)-1\bigr)t^{\frac{s}{2}-1}\,\mathrm{d}t,</math>

in terms of Jacobi's theta function

<math>\theta(\tau)= \sum_{n=-\infty}^\infty e^{\pi i n^2\tau}.</math>

However, this integral only converges if the real part of Template:Math is greater than Template:Math, but it can be regularized. This gives the following expression for the zeta function, which is well defined for all Template:Math except Template:Math and Template:Math:

<math> \pi^{-\frac{s}{2}}\Gamma\left(\frac{s}{2}\right)\zeta(s) = \frac{1}{s-1}-\frac{1}{s} +\frac{1}{2} \int_0^1 \left(\theta(it)-t^{-\frac12}\right)t^{\frac{s}{2}-1}\,\mathrm{d}t + \frac{1}{2}\int_1^\infty \bigl(\theta(it)-1\bigr)t^{\frac{s}{2}-1}\,\mathrm{d}t.</math>

Laurent series

The Riemann zeta function is meromorphic with a single pole of order one at Template:Math. It can therefore be expanded as a Laurent series about Template:Math; the series development is then<ref>Template:Cite journal</ref>

<math>\zeta(s)=\frac{1}{s-1}+\sum_{n=0}^\infty \frac{\gamma_n}{n!}(1-s)^n.</math>

The constants Template:Math here are called the Stieltjes constants and can be defined by the limit

<math> \gamma_n = \lim_{m \rightarrow \infty}{\left(\left(\sum_{k = 1}^m \frac{(\ln k)^n}{k}\right) - \frac{(\ln m)^{n+1}}{n+1}\right)}.</math>

The constant term Template:Math is the Euler–Mascheroni constant.

Integral

For all Template:Math, Template:Math, the integral relation (cf. Abel–Plana formula)

<math>\zeta(s) = \frac{ 1 }{s - 1} + \frac{1}{2} + 2 \int_0^{\infty} \frac{\sin(s\arctan t) }{ \left(1 + t^2 \right)^{s/2} \left(e^{2\pi t} - 1\right)\ }\ \operatorname{d}t </math>

holds true, which may be used for a numerical evaluation of the zeta function.

Hadamard product

On the basis of Weierstrass's factorization theorem, Hadamard gave the infinite product expansion

<math>\zeta(s) = \frac{e^{\left(\log(2\pi)-1-\frac{\gamma}{2}\right)s}}{2(s-1)\Gamma\left(1+\frac{s}{2}\right)} \prod_\rho \left(1 - \frac{s}{\rho} \right) e^\frac{s}{\rho},</math>

where the product is over the non-trivial zeros Template:Math of Template:Math and the letter Template:Math again denotes the Euler–Mascheroni constant. A simpler infinite product expansion is

<math>\zeta(s) = \pi^\frac{s}{2} \frac{\prod_\rho \left(1 - \frac{s}{\rho} \right)}{2(s-1)\Gamma\left(1+\frac{s}{2}\right)}.</math>

This form clearly displays the simple pole at Template:Math, the trivial zeros at Template:Math... due to the gamma function term in the denominator, and the non-trivial zeros at Template:Math. (To ensure convergence in the latter formula, the product should be taken over "matching pairs" of zeros, i.e. the factors for a pair of zeros of the form Template:Math and Template:Math should be combined.)

Globally convergent series

A globally convergent series for the zeta function, valid for all complex numbers Template:Math except Template:Math for some integer Template:Math, was conjectured by Konrad Knopp in 1926 <ref name="blag2018" /> and proven by Helmut Hasse in 1930<ref name = Hasse1930 /> (cf. Euler summation):

<math>\zeta(s)=\frac{1}{1-2^{1-s}} \sum_{n=0}^\infty \frac {1}{2^{n+1}} \sum_{k=0}^n \binom{n}{k} \frac{(-1)^k}{(k+1)^{s}}.</math>

The series appeared in an appendix to Hasse's paper, and was published for the second time by Jonathan Sondow in 1994.<ref>Template:Cite journal</ref>

Hasse also proved the globally converging series

<math>\zeta(s)=\frac 1{s-1}\sum_{n=0}^\infty \frac 1{n+1}\sum_{k=0}^n\binom {n}{k}\frac{(-1)^k}{(k+1)^{s-1}}</math>

in the same publication.<ref name = Hasse1930 /> Research by Iaroslav Blagouchine<ref> Template:Cite journal</ref><ref name="blag2018"> Template:Cite journal</ref> has found that a similar, equivalent series was published by Joseph Ser in 1926.<ref>Template:Cite journal</ref>

In 1997 K. Maślanka gave another globally convergent (except Template:Math) series for the Riemann zeta function:

<math>\zeta (s)=\frac{1}{s-1}\sum_{k=0}^\infty \biggl(\prod_{i=1}^{k} (i-\frac{s}{2})\biggl) \frac{A_{k}}{k!}=

\frac{1}{s-1} \sum_{k=0}^\infty \biggl(1-\frac{s}{2}\biggl)_{k} \frac{A_{k}}{k!}</math> where real coefficients <math>A_k</math> are given by:

<math>A_k=\sum_{j=0}^{k}(-1)^{j}\binom{k}{j}(2j+1)\zeta

(2j+2)=\sum_{j=0}^{k}\binom{k}{j}\frac{B_{2j+2}\pi ^{2j+2}}{\left(2\right) _{j}\left( \frac{1}{2}\right) _{j}} </math>

Here Template:Math are the Bernoulli numbers and Template:Math denotes the Pochhammer symbol.<ref>Template:Cite journal</ref><ref> Template:Cite journal</ref>

Note that this representation of the zeta function is essentially an interpolation with nodes, where the nodes are points Template:Math, i.e. exactly those where the zeta values are precisely known, as Euler showed. An elegant and very short proof of this representation of the zeta function, based on Carlson's theorem, was presented by Philippe Flajolet in 2006.<ref>Template:Cite journal</ref>

The asymptotic behavior of the coefficients <math>A_{k}</math> is rather curious: for growing <math>k</math> values, we observe regular oscillations with a nearly exponentially decreasing amplitude and slowly decreasing frequency (roughly as <math>k^{-2/3}</math>). Using the saddle point method, we can show that

<math>A_{k}\sim \frac{4\pi ^{3/2}}{\sqrt{3\kappa }}\exp \biggl( -\frac{3\kappa }{2}+\frac{\pi ^{2}}{4\kappa }\biggl) \cos \biggl( \frac{4\pi }{3}-\frac{3\sqrt{3}

\kappa }{2}+\frac{\sqrt{3}\pi ^{2}}{4\kappa }\biggl)</math> where <math>\kappa</math> stands for:

<math>\kappa :=\sqrt[3]{\pi ^{2}k} </math>

(see <ref>Template:Cite journal</ref> for details).

On the basis of this representation, in 2003 Luis Báez-Duarte provided a new criterion for the Riemann hypothesis.<ref>Template:Cite journal</ref><ref> Template:Cite journal</ref><ref> Template:Cite journal</ref> Namely, if we define the coefficients Template:Math as

<math>c_{k}:=\sum_{j=0}^{k}(-1)^{j}\binom{k}{j}\frac{1}{\zeta (2j+2)}</math>

then the Riemann hypothesis is equivalent to

<math>c_{k}=\mathcal{O}\left( k^{-3/4+\varepsilon }\right) \qquad (\forall\varepsilon >0) </math>

Rapidly convergent series

Peter Borwein 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.<ref>Template:Cite book</ref>

Series representation at positive integers via the primorial

<math> \zeta(k)=\frac{2^k}{2^k-1}+\sum_{r=2}^\infty\frac{(p_{r-1}\#)^k}{J_k(p_r\#)}\qquad k=2,3,\ldots.</math>

Here Template:Math is the primorial sequence and Template:Math is Jordan's totient function.<ref>Template:Cite journal</ref>

Series representation by the incomplete poly-Bernoulli numbers

The function Template:Math can be represented, for Template:Math, by the infinite series

<math>\zeta(s)=\sum_{n=0}^\infty B_{n,\ge2}^{(s)}\frac{(W_k(-1))^n}{n!},</math>

where Template:Math, Template:Math is the Template:Mvarth branch of the [[Lambert W function|Lambert Template:Mvar-function]], and Template:Math is an incomplete poly-Bernoulli number.<ref>Template:Cite journal</ref>

Mellin transform of the Engel map

The function Template:Math is iterated to find the coefficients appearing in Engel expansions.<ref>Template:Cite web</ref>

The Mellin transform of the map <math>g(x)</math> is related to the Riemann zeta function by the formula

<math> \begin{align}
   \int_0^1 g (x) x^{s - 1} \, dx & = \sum_{n = 1}^\infty
   \int_{\frac{1}{n + 1}}^{\frac{1}{n}} (x (n + 1) - 1) x^{s - 1} \, d x\\[6pt]
   & = \sum_{n = 1}^\infty \frac{n^{- s} (s - 1) + (n + 1)^{- s - 1} (n^2 + 2 n + 1) + n^{- s - 1} s - n^{1 - s}}{(s + 1) s (n + 1)}\\[6pt]
   & = \frac{\zeta (s + 1)}{s + 1} - \frac{1}{s (s + 1)}
 \end{align}</math>

Stochastic representations

The Brownian motion and Riemann zeta function are connected through the moment-generating functions of stochastic processes derived from the Brownian motion.<ref>Template:Cite journal</ref>

Numerical algorithms

A classical algorithm, in use prior to about 1930, proceeds by applying the Euler–Maclaurin formula to obtain, for positive integers Template:Math and Template:Math,

<math>\zeta(s) = \sum_{j=1}^{n-1}j^{-s} + \tfrac12 n^{-s} + \frac{n^{1-s}}{s-1} + \sum_{k=1}^m T_{k,n}(s) + E_{m,n}(s)</math>

where, letting <math>B_{2k}</math> denote the indicated Bernoulli number,

<math>T_{k,n}(s) = \frac{B_{2k}}{(2k)!} n^{1-s-2k}\prod_{j=0}^{2k-2}(s+j)</math>

and the error satisfies

<math>|E_{m,n}(s)| < \left|\frac{s+2m+1}{\sigma + 2m + 1}T_{m+1,n}(s)\right|,</math>

with Template:Math.<ref>Template:Cite journal </ref>

A modern numerical algorithm is the Odlyzko–Schönhage algorithm.

Applications

The zeta function occurs in applied statistics including Zipf's law, Zipf–Mandelbrot law, and Lotka's law.

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.<ref>Template:Cite web</ref>

Musical tuning

In the theory of musical tunings, the zeta function can be used to find equal divisions of the octave (EDOs) that closely approximate the intervals of the harmonic series. For increasing values of <math>t \in \mathbb{R}</math>, the value of

<math>\left\vert \zeta \left( \frac{1}{2} + \frac{2\pi{i}}{\ln{(2)}}t \right) \right\vert</math>

peaks near integers that correspond to such EDOs.<ref>Template:Cite web</ref> Examples include popular choices such as 12, 19, and 53.<ref>Template:Cite book</ref>

Infinite series

The zeta function evaluated at equidistant positive integers appears in infinite series representations of a number of constants.<ref>Most of the formulas in this section are from § 4 of J. M. Borwein et al. (2000)</ref>

  • <math>\sum_{n=2}^\infty\bigl(\zeta(n)-1\bigr) = 1</math>

In fact the even and odd terms give the two sums

  • <math>\sum_{n=1}^\infty\bigl(\zeta(2n)-1\bigr)=\frac{3}{4}</math>

and

  • <math>\sum_{n=1}^\infty\bigl(\zeta(2n+1)-1\bigr)=\frac{1}{4}</math>

Parametrized versions of the above sums are given by

  • <math>\sum_{n=1}^\infty(\zeta(2n)-1)\,t^{2n} = \frac{t^2}{t^2-1} + \frac{1}{2} \left(1- \pi t\cot(t\pi)\right)</math>

and

  • <math>\sum_{n=1}^\infty(\zeta(2n+1)-1)\,t^{2n} = \frac{t^2}{t^2-1} -\frac{1}{2}\left(\psi^0(t)+\psi^0(-t) \right) - \gamma</math>

with Template:Math and where <math>\psi</math> and <math>\gamma</math> are the polygamma function and Euler's constant, respectively, as well as

  • <math>\sum_{n=1}^\infty \frac{\zeta(2n)-1}{n}\,t^{2n} = \log\left(\dfrac{1-t^2}{\operatorname{sinc}(\pi\,t)}\right)</math>

all of which are continuous at <math>t=1</math>. Other sums include

  • <math>\sum_{n=2}^\infty\frac{\zeta(n)-1}{n} = 1-\gamma</math>
  • <math>\sum_{n=1}^\infty\frac{\zeta(2n)-1}{n} = \ln 2</math>
  • <math>\sum_{n=2}^\infty\frac{\zeta(n)-1}{n} \left(\left(\tfrac{3}{2}\right)^{n-1}-1\right) = \frac{1}{3} \ln \pi</math>
  • <math>\sum_{n=1}^\infty\bigl(\zeta(4n)-1\bigr) = \frac78-\frac{\pi}{4}\left(\frac{e^{2\pi}+1}{e^{2\pi}-1}\right)</math>
  • <math>\sum_{n=2}^\infty\frac{\zeta(n)-1}{n}\Im \bigl((1+i)^n-1-i^n\bigr) = \frac{\pi}{4}</math>

where <math>\Im</math> denotes the imaginary part of a complex number.

Another interesting series that relates to the natural logarithm of the lemniscate constant is the following

  • <math>\sum_{n=2}^\infty\left[\frac{2(-1)^n\zeta(n)}{4^n n}-\frac{(-1)^n\zeta(n)}{2^n n} \right]= \ln \left( \frac{\varpi}{2\sqrt2} \right)

</math>

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

<math>\zeta(s,q) = \sum_{k=0}^\infty \frac{1}{(k+q)^s}</math>

(the convergent series representation was given by Helmut Hasse in 1930,<ref name = Hasse1930>Template:Cite journal</ref> cf. Hurwitz zeta function), which coincides with the Riemann zeta function when Template:Math (the lower limit of summation in the Hurwitz zeta function is Template:Math, not Template:Math), the [[Dirichlet L-function|Dirichlet Template:Math-functions]] and the Dedekind zeta function. For other related functions see the articles zeta function and [[L-function|Template:Math-function]].

The polylogarithm is given by

<math>\operatorname{Li}_s(z) = \sum_{k=1}^\infty \frac{z^k}{k^s}</math>

which coincides with the Riemann zeta function when Template:Math. The Clausen function Template:Math can be chosen as the real or imaginary part of Template:Math.

The Lerch transcendent is given by

<math>\Phi(z, s, q) = \sum_{k=0}^\infty\frac {z^k} {(k+q)^s}</math>

which coincides with the Riemann zeta function when Template:Math and Template:Math (the lower limit of summation in the Lerch transcendent is Template:Math, not Template:Math).

The multiple zeta functions are defined by

<math>\zeta(s_1,s_2,\ldots,s_n) = \sum_{k_1>k_2>\cdots>k_n>0} {k_1}^{-s_1}{k_2}^{-s_2}\cdots {k_n}^{-s_n}.</math>

One can analytically continue these functions to the Template:Math-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.

See also

References

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Sources

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