Function zeta of Riemann

The function zeta of Riemann $\backslash \; zeta$ is a definite Fonction méromorphe for all Complex number $s$ of real part $\backslash \; Re\; E\; \backslash \; (S)\; >\; 1$ by the convergent series:

Defined initially by Leonhard Euler which considers only the actual values of S, it is prolonged in a function méromorphe admitting a pole of residue 1 in S = 1 (Theorem of Dirichlet). Bernhard Riemann realized that the function $\backslash \; zeta$ can be prolonged analytically (in a single way) with all the complex numbers different from 1.

It is a almost periodic Fonction within the meaning of Harald Bohr in the half-plane $\backslash \; Re\; E\; \backslash \; (S)\; >\; 1$

It is this function which is the subject of the Hypothèse of Riemann.

For the study of the series itself (convergence, sums partial), to see the article Series of Riemann.

Functional equation

The function $\backslash \; zeta$ satisfy the functional equation:

valid for any complex number $s$ different from 0 and 1. Here, $\backslash \; Gamma$ indicates the function Gamma. This formula is used to build the analytical prolongation. In $s\; =\; 1$, the function $\backslash \; zeta$ has a simple pole of residue equal to 1.

Proof of the functional equation

Relations for the function $\backslash \; Pi$

See also: Function gamma

The factorial function can be wide with the whole of the complex numbers of real part strictly higher than -1 by:

$\backslash \; pi\; (S)\; =\; \backslash \; int\_\; \{0\}\; ^\; \{+\; \backslash \; infty\}\; e^\; \{-\; X\}\; x^s\; \backslash \; mathrm\; \{D\}\; X\; \backslash \; qquad\; (1)$

for $s$ complex such as $\backslash \; mathrm\; \{Re\}\; (S)\; >-1$.

Indeed, for entire naturalness $n$ one obtains: $\backslash \; pi\; (N)\; =\; N!$

One can easily make sure of the exactitude of the equality with an integration by parts in the first formula. At the end there remains only the nth derivative of $x^n$ compared to $x$, which is worth $n\; of\; course!$.

Relationship with the function zeta

That is to say $s$ a complex number which has to some extent real a reality strictly higher than 1. Starting from the equation (1), in substituent $nx$ with $x$ with $n\; \backslash \; in\; \backslash \; mathbb\; \{NR\}\; ^*$, one can obtain the following relation:

$$

\ frac {\ pi (s-1)}{n^s} = \ int_ {0} ^ {+ \ infty} e^ {- nx} x^ {s-1} \ mathrm {D} x Then while summoning on N (there is absolute Convergence well because $\backslash \; mathrm\; \{Re\}\; (S)\; >1$, to see on this subject the article Série of Riemann):

$\backslash \; pi\; (s-1)\; \backslash \; sum\_\; \{n=1\}\; ^\; \{+\; \backslash \; infty\}\; \backslash \; frac\; \{1\}\; \{n^s\}\; =\; \backslash \; int\_\; \{0\}\; ^\; \{+\; \backslash \; infty\}\; \backslash \; sum\_\; \{n=1\}\; ^\; \{+\; \backslash \; infty\}\; e^\; \{-\; nx\}\; x^\; \{s-1\}\; \backslash \; mathrm\; \{D\}\; X\; =\; \backslash \; int\_\; \{0\}\; ^\; \{+\; \backslash \; infty\}\; \backslash \; frac\; \{x^\; \{s-1\}\}\; \{e^x\; -\; 1\}\; \backslash \; mathrm\; \{D\}\; X\; \backslash \; qquad\; \backslash \; mathrm\; \{because\}\; \backslash \; qquad\; \backslash \; sum\_\; \{n=1\}\; ^\; \{+\; \backslash \; infty\}\; r^\; \{-\; N\}\; =\; \backslash \; frac\; \{1\}\; \{r-1\}$

\ qquad (2)

Transformation of the integral

Then it is necessary to consider the following integral of contour:

$\backslash \; int\_\; \{+\; \backslash \; infty\}\; ^\; \{+\; \backslash \; infty\}\; \backslash \; frac\; \{(-\; X)\; ^\; \{S\}\}\; \{e^x\; -\; 1\}\; \backslash \; frac\; \{\backslash \; mathrm\; \{D\}\; X\}\; \{X\}\; =\; \backslash \; int\_\; \{+\; \backslash \; infty\}\; ^\; \{\backslash \; delta\}\; \backslash \; frac\; \{(-\; X)\; ^\; \{S\}\}\; \{e^x\; -\; 1\}\; \backslash \; frac\; \{\backslash \; mathrm\; \{D\}\; X\}\; \{X\}\; \backslash \; mathrm\; \{D\}\; x+\; \backslash \; int\_\; \{\backslash \; green\; X\; \backslash \; green\; =\; \backslash \; delta\}\; \backslash \; frac\; \{(-\; X)\; ^\; \{S\}\}\; \{e^x\; -\; 1\}\; \backslash \; frac\; \{\backslash \; mathrm\; \{D\}\; X\}\; \{X\}$

+ \ int_ {\ delta} ^ {+ \ infty} \ frac {(- X) ^ {S}} {e^x - 1} \ frac {\ mathrm {D} X} {X}

The way of integration comes from infinite until $\backslash \; delta$, by above the axis of realities, turns around the origin and sets out again towards the infinite one by below the axis of realities. Initially let us evaluate the term of the center by introducing polar coordinates:

$x\; =\; \backslash \; green\; X\; \backslash \; green\; \backslash \; cos\; (\backslash \; theta)\; +\; I\; \backslash \; green\; X\; \backslash \; green\; \backslash \; sin\; (\backslash \; theta)\; \backslash \; qquad\; \backslash \; mathrm\; \{thus\}\; \backslash \; qquad\; \backslash \; frac\; \{\backslash \; mathrm\; \{D\}\; X\}\; \{\backslash \; mathrm\; \{D\}\; \backslash \; theta\}\; =\; -\; \backslash \; green\; X\; \backslash \; green\; \backslash \; cos\; (\backslash \; theta)\; +\; I\; \backslash \; green\; X\; \backslash \; green\; \backslash \; cos\; (\backslash \; theta)\; =\; I\; X\; \backslash \; qquad\; \backslash \; mathrm\; \{thus\}\; \backslash \; qquad\; \backslash \; frac\; \{\backslash \; mathrm\; \{D\}\; X\}\; \{X\}\; =\; I\; \backslash \; mathrm\; \{D\}\; \backslash \; theta$

Like $-x/(e^x\; -\; 1)\; =\; -1$, the term of the center is worth simply 0 when $\backslash \; delta\; \backslash \; to\; 0$ and that $\backslash \; mathrm\; \{Re\}\; (S)\; >\; 1$.

The two other terms can be written like:

Expression for zeta

The equality (2) becomes now:

$\backslash \; int\_\; \{+\; \backslash \; infty\}\; ^\; \{+\; \backslash \; infty\}\; \backslash \; frac\; \{(-\; X)\; ^\; \{S\}\}\; \{e^x\; -\; 1\}\; \backslash \; frac\; \{\backslash \; mathrm\; \{D\}\; X\}\; \{X\}\; =\; 2\; I\; \backslash \; sin\; (\backslash \; pi\; S)\; \backslash \; pi\; (s-1)\; \backslash \; sum\_\; \{n=1\}\; ^\; \backslash \; infty\; \backslash \; frac\; \{1\}\; \{n^s\}$

By using a relation for the function $\backslash \; Pi$ one can isolate $\backslash \; zeta$:

$\backslash \; zeta\; (S)\; =\; \backslash \; frac\; \{\backslash \; pi\; (-\; S)\}\{2\; \backslash \; pi\; I\}\; \backslash \; int\_\; \{+\; \backslash \; infty\}\; ^\; \{+\; \backslash \; infty\}\; \backslash \; frac\; \{(-\; X)\; ^\; \{S\}\}\; \{e^x\; -\; 1\}\; \backslash \; frac\; \{\backslash \; mathrm\; \{D\}\; X\}\; \{X\}\; \backslash \; quad\; \backslash \; mathrm\; \{because\}\; \backslash \; quad\; \backslash \; pi\; S\; =\; \backslash \; pi\; (S)\; \backslash \; pi\; (-\; S)\; \backslash \; sin\; (\backslash \; pi\; S)\; \backslash \; qquad\; (3)$

Now it is a question of estimating the integral; in the plan complexes the function to integrate is everywhere defined except:

• On the positive part of the axis of realities since $(-\; X)\; ^s\; =\; e^\; \{S\; \backslash \; log\; (-\; X)\}$ is not there défini
  
• At the points $x\; =\; \backslash \; pm\; 2\; \backslash \; pi\; I\; N\; \backslash \; qquad\; N\; \backslash \; in\; \backslash \; mathbb\; N^*$

According to the integral Theorem of Cauchy:

$\backslash \; int\_\; \{\backslash \; partial\; D\}\; \backslash \; frac\; \{(-\; X)\; ^\; \{S\}\}\; \{e^x\; -\; 1\}\; \backslash \; frac\; \{\backslash \; mathrm\; \{D\}\; X\}\; \{X\}\; =\; 0$ where $D$ is the field of

definition

By posing $x\; =\; 2\; \backslash \; pi\; I\; N\; +\; y$ where $\backslash \; green\; there\; \backslash \; green\; =\; \backslash \; varepsilon$:

$\backslash \; int\_\; \{+\; \backslash \; infty\}\; ^\; \{+\; \backslash \; infty\}\; \backslash \; frac\; \{(-\; X)\; ^\; \{S\}\}\; \{e^x\; -\; 1\}\; \backslash \; frac\; \{\backslash \; mathrm\; \{D\}\; X\}\; \{X\}\; +\; \backslash \; sum\_\; \{n=1\}\; ^\; \backslash \; infty\; \backslash \; int\_\; \{\backslash \; green\; there\; \backslash \; green\; =\; \backslash \; varepsilon\}\; (\backslash \; pm\; 2\; \backslash \; pi\; I\; N\; +\; there)\; ^\; \{s-1\}\; \backslash \; frac\; \{there\}\; \{e^y\; -\; 1\}\; \backslash \; frac\; \{\backslash \; mathrm\; \{D\}\; there\}\; \{there\}\; =\; 0\; \backslash \; qquad\; (4)$

For the second term there is now $y/(e^y\; -\; 1)\; =\; 1$ when $\backslash \; varepsilon\; \backslash \; to\; 0$ and thus the integral becomes:

$\backslash \; int\_\; \{\backslash \; green\; there\; \backslash \; green\; =\; \backslash \; varepsilon\}\; \backslash \; frac\; \{F\; (there)\}\{there\; -\; 0\}\; \backslash \; mathrm\; \{D\}\; there\; \backslash \; quad\; \backslash \; mathrm\; \{O\; \backslash \; serious\; \{U\}\}\; \backslash \; quad\; F\; (there)\; =\; (\backslash \; pm\; 2\; \backslash \; pi\; I\; N\; +\; there)\; ^\; \{s-1\}$

By applying the integral Formule of Cauchy one finds that the integral is worth $2\; \backslash \; pi\; I\; F\; (0)$, therefore the expression (4) becomes:

$\backslash \; begin\; \{array\}\; \{L\}$

\ displaystyle \ int_ {+ \ infty} ^ {+ \ infty} \ frac {(- X) ^ {S}} {e^x - 1} \ frac {\ mathrm {D} X} {X} & = \ displaystyle - 2 \ pi I \ sum_ {n=1} ^ \ infty (- 2 \ pi I N) ^ {s-1} + (2 \ pi I N) ^ {s-1} = - 2 \ pi I (2 \ pi) ^ {s-1} \ sum_ {n=1} ^ \ infty \ frac {1} {n^ {1-s}}\ \ & \ \ & \ displaystyle = - 2 \ pi I (2 \ pi) ^ {s-1} e^ {- I (\ pi/2) (s-1)} + e^ {I (\ pi/2) (s-1)} \ zeta (1-s) \ \ & \ \ & \ displaystyle = - (2 \ pi) ^ {S} I e^ {- I (\ pi/2) S} - I e^ {I (\ pi/2) S)} \ zeta (1-s) \ \ & \ \ & \ displaystyle = (2 \ pi) ^ {S} (- 2 I) \ sin (\ frac {S \ pi} {2}) \ zeta (1-s) \ end {array}

Finally by replacing the integral in the expression (3) by his value one finds the equation functional:

$\backslash \; zeta\; (S)\; =\; \backslash \; frac\; \{\backslash \; pi\; (-\; S)\}\{2\; \backslash \; pi\; I\}\; (2\; \backslash \; pi)\; ^\; \{S\}\; (-\; 2\; I)\; \backslash \; sin\; (\backslash \; frac\; \{S\; \backslash \; pi\}\; \{2\})\; \backslash \; zeta\; (1-s)\; =\; -\; 2\; \backslash \; pi\; (-\; S)\; (2\; \backslash \; pi)\; ^\; \{s-1\}\; \backslash \; sin\; (\backslash \; frac\; \{S\; \backslash \; pi\}\; \{2\})\; \backslash \; zeta\; (1-s)\; \backslash \; qquad\; (5)$

i.e.

$\backslash \; zeta\; (S)\; =2^s\; \backslash \; pi^\; \{s-1\}\; \backslash \; pi\; (-\; S)\; \backslash \; zeta\; (s-1)$

However the definition of the Function gamma, for $z\; \backslash \; in\; \backslash \; mathbb\; \{C\}$ such as $\backslash \; mathrm\; \{Re\}\; (Z)\; >0$, is

$\backslash \; Gamma:\; Z\; \backslash \; longmapsto\; \backslash \; int\_0^\; \{+\; \backslash \; infty\}\; t^\; \{z-1\}\; e^\; \{-\; T\}\; \backslash \; mathrm\; \{D\}\; t$

Thus one a:

$\backslash \; pi\; (Z)\; =\; \backslash \; Gamma\; (z+1)\; \backslash \; quad\; \backslash \; mathrm\; \{thus\}\; \backslash \; quad\; \backslash \; pi\; (-\; S)\; =\; \backslash \; Gamma\; (1-s)$

And finally while using the relation (5) it comes:

$\backslash \; zeta\; (S)\; =2^s\; \backslash \; pi^\; \{s-1\}\; \backslash \; sin\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; pi\; S\}\; \{2\}\; \backslash \; right)\; \backslash \; Gamma\; (1-s)\; \backslash \; zeta\; (1-s)$

what completes the demonstration.

Bond with the prime numbers

The bond between the function $\backslash \; zeta$ and the prime numbers had already been established by Leonhard Euler with the formula:

where the infinite Produit is extended to the unit $\backslash \; mathcal\; P$ of the prime numbers. This relation is a consequence of the formula for the geometrical continuations and of the fundamental Théorème of arithmetic the. One calls sometimes this formula Produit eulérien.

Zeros of the function zeta

Zero commonplace

The function $\backslash \; zeta$ presents a continuation of zero known as commonplace on the negative real axis:

To be convinced some, it is enough to look at the functional equation of the function zeta, the multiple pars of negative entireties cancelling the sine. These zeros all are simple. The functional relation makes it possible to calculate the approximation with the first order of the function zeta in the vicinity of the entirety -2p.

Zero noncommonplace - Assumption of Riemann

Famous the Hypothèse of Riemann is a conjecture formulated in 1859 by the mathematician Bernhard Riemann. She says that the zero noncommonplace $s\_k$ of the function $\backslash \; zeta$ have all to some extent real 1/2, i.e.:

Three things are established:

• the function zeta admits an infinity of zeros on the axis $\backslash \; Re\; E\; (S)\; =1/2$ (theorem of Hardy, 1914)

• There does not exist zero on the axis $\backslash \; Re\; \{E\}\; (S)\; =1$. This statement is equivalent to the Théorème of the prime numbers, which is due independently to Jacques Hadamard and Charles-Jean de la Vallee poussin in 1896. Equivalence with the theorem of the prime numbers is due to Landau.

• the function zeta of Riemann is not cancelled for $s$ pertaining to the area defined by

where T indicates the imaginary part of S and C a suitable constant. This result was proven by Vinogradov and Korobov in 1958.

Hilbert and Polya suggested that the conjecture of Riemann would be shown if one could find an operator Hermitien $\backslash \; hat\; \{H\}$ of which the eigenvalues (necessarily real) are exactly the imaginary parts $E\_k$ of zero the noncommonplace ones:

Such a square operator was not found yet explicitly to date. Nevertheless, this equation with the eigenvalues suggests a bond with a problem of quantum Mécanique not relativist who is specified in the following paragraph.

History

See also: History of the function Zeta of Riemann

Properties

Bonds with the prime numbers

The zeros of the function $\backslash \; zeta$ play a big role, because certain integrals of contour implying the function $s\; \backslash \; mapsto\; \backslash \; ln\; (1\; \backslash \; zeta\; (S))$ can be used to approach the function π number of prime numbers (see Théorème of the prime numbers). These integrals of contour are calculated by means of the Remainder theorem, and the knowledge of the singularities is thus necessary.

Statistical properties of zero the noncommonplace ones & quantum chaos

The statistical properties of zero the noncommonplace ones of the function $\backslash \; zeta$ resemble asymptotically that of the eigenvalues of a great whole of random Matrices unit Gaussian of the FORD unit. This conjecture is based on many numerical results, and strongly supported by a rigorous theorem of Montgomery. This led the physicist theorist Michael Berry to conjecture that the imaginary parts $E\_k$ of zero the noncommonplace ones could be interpreted as the eigenvalues of an operator Hamiltonien describing a quantum system not relativist who would be classically chaotic, and whose traditional orbits do not have the symmetry of inversion of the temps^{, ,} . Better, a Hamiltonian operator pretense to have the good properties was recently exhibé by Berry and Keating^, .

The statistical properties of zero the noncommonplace ones continue to be the object of intense research, as well numerical as analytical. One will be able to also read: Philippe Biane; the function zeta of Riemann and the probabilities , Days X-UPS (2003), text with format pdf.

Various aspects

• Euler was able to calculate the value of the function $\backslash \; zeta\; \backslash ,$ for the whole positive even by using the formula:

valid for entire positive $k$, where the $B\_\; \{2k\}\; \backslash ,$ are the Nombres of Bernoulli. From there, we see that:

We obtain the famous infinite series allowing to calculate the even powers of π. For the odd entireties, the case is not so simple. Ramanujan worked much on these series and Apéry showed in 1979 that $\backslash \; zeta\; (3)\; =\; 1,2020569\dots \; \backslash ,$ is irrational (see Constante of Apéry). In 2000, Tanguy Rivoal showed that there exists an infinity of irrational numbers among the values with the odd entireties. One conjectures that all the values with the odd entireties are irrational and even transcendent (transcendent Nombre).

• One can express the reverse of the function $\backslash \; zeta\; \backslash ,$ by using the Fonction of Möbius $\backslash \; driven\; \backslash ,$ using the formula:

valid for any complex number $s$ of real part $\backslash \; Re\; E\; \backslash \; (S)\; >\; 1\; \backslash ,$. This formula, jointly with the expression of $\backslash \; zeta\; (2)\; \backslash ,$ given higher, can be used to show that the probability so that two integers taken randomly are Premiers between them is equal to $\backslash \; frac\; \{6\}\; \{\backslash \; pi^2\}\; \backslash ,$, that is to say approximately 60,8 percent.

See too

References

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