History of the function Zeta of Riemann

Work of Legendre

• Euler was not going to be long in being contradicted. When one is confronted with a function having variations which seem anarchistic, the first idea which comes is to try to smooth these data. Such a smoothing can be an average, possibly mobile, but one can also be interested in the sum of these values or the simple counting of the number of terms in a given interval. It is the initial idea of Adrien-Marie Legendre which since 1785 seeks a formula approached for the number of the prime numbers smaller than $x\; \backslash ,$, than it notes $\backslash \; pi\; (X)\; \backslash ,$ and than one calls today the Fonction of account of the prime numbers. And he proposes the formula

for two constant, and B has chosen well.

And, by an argument Heuristique, he conjectures that the number of prime number inferiors or equal to $x\; \backslash ,$ and contained in the arithmetic progression $a.n+b\; \backslash ,$, with $a\; \backslash ,$ and $b\; \backslash ,$ First between them, is $\backslash \; pi\; (X,\; has,\; b)\; \backslash \; approx\; \backslash \; frac\; \{1\}\; \{\backslash \; varphi\; (a)\}\; \backslash \; pi\; (X)\; \backslash ,$ where $\backslash \; varphi\; (K)\; \backslash ,$ is the number of the entireties first with $k\; \backslash ,$ and lower than $k\; \backslash ,$.

• Legendre publishes in its Essai on the theory of the numbers the first form of what will become the Théorème of the prime numbers, shown independently by Jacques Hadamard and Charles-Jean de la Vallee poussin in 1896.

Number 1,08366 since is called number of Legendre .

• Thereafter, in a letter with Encke gone back to 1849, Gauss will support to have since 1793 a conjecture of the same type. But Gauss did not publish anything alive sound on this question. Gauss claims to have noted on the table of the prime numbers that the Probabilité that an entirety N (odd!) either first is approximately $1/\backslash \; ln\; n$. Extremely of this conjecture, it from of naturally deduced the formula

• Using a process close to the formula of the screen (which thus bears the name of screen of Eratosthène-Legendre), Legendre finds finally the Formule of Legendre (1808)

• Legendre deduced this first result, new from it since antiquity, on the distribution of the prime numbers

  $\backslash \; lim\_\; \{X\; \backslash \; rightarrow\; \backslash \; infty\}\; \{\backslash \; frac\; \{\backslash \; pi\; (X)\}\{X\}\}\; =0.$ Donc the Proportion of the prime numbers tends towards $0\; \backslash ,$. This natural impression thus is found that the prime numbers are increasingly rare as one goes further in the list. This theorem is called theorem of rarefaction of the prime numbers.

• In its theory of the numbers, emitted Legendre a conjecture, which bears now the name of conjecture of Legendre , and which states that there exists a prime number p ranging between N ² and (n+1) ² for entire N. This conjecture is related to the assumption of Riemann in the following way:

Let us take for N the value $$. According to the conjecture there would exist a prime number p between N ² and (n+1) ². There are thus the inequalities

that is to say still, since $p\_\; \{m+1\}\; \backslash \; the\; p$,

It will be seen that the assumption of Riemann implies for constant C > 0 adapted

Series of Dirichlet

Taking again the demonstration of Euler on the infinitude of the prime numbers, Dirichlet arrives between 1837 and 1839 have to show a consequence of a conjecture of Legendre going back to 1785.

Theorem of Dirichlet:

“If the $a$ numbers and $b$ are first between them, there exists an infinity of prime numbers in the arithmetic progression $an+b$, $n\; \backslash \; in\; \backslash \; mathbb\; \{NR\}$. ”

But for that it will associate a series, which one calls since Série of Dirichlet, and which is form

Taking again the argument of the demonstration of Euler on the infinity of the prime numbers and the divergence of the series of the opposite of the prime numbers, he deduced his theorem from it from the presence of a pole of the associated series in s=1.

He showed that one had

The postulate of Bertrand

By analyzing a table of prime numbers up to 6.000.000, Joseph Bertrand states conjecture:

Postulate of Bertrand (1845): “Between N and 2n exists always a prime number. ”

It is with the demonstration of this result that will work Tchebyscheff.

Work of Tchebyscheff

In 1849, Tchebyscheff shows that if $\backslash \; pi\; (X)\; \backslash \; ln\; (X)\; /x$ tends towards a limit, the limit is equal to 1. Then in 1850, Tchebyscheff, using the formula of Stirling astutely shows a weak form of the conjecture of Legendre,

Theorem: 'For $x\; \backslash \; Ge\; 30$, there is $A\; \backslash \; frac\; \{X\}\; \{\backslash \; ln\; X\}\; \backslash \; \backslash \; pi\; (X)\; \backslash \; the\; \backslash \; frac\; \{6\}\; \{5\}\; has\; \backslash \; frac\; \{X\}\; \{\backslash \; ln\; X\}$ with $A=\; \backslash \; ln\; \{\backslash \; frac\; \{2^\; \{1/2\}\; 3^\; \{1/3\}\; 4^\; \{1/4\}\}\; \{30^\; \{1/30\}\}\}\; \backslash \; approx\; 0.92.$

and from of deduced the postulate from Bertrand. But it is unable to show the existence of the limit.

He benefits from it to show that the number from Legendre, 1,08366, must be replaced by 1.

These results will have a considerable influence. It should here be remembered that to make mathematics until the XIXe century is to calculate on equalities. One sees here appearing inequalities, thing well not very current whereas they are currencies at our time.

The childhood of zeta

The report of Riemann

• the beginning of the XIXe century saw creating the theory of the complex analytical functions and the methods of the modern analysis. Cauchy discovers the Remainder theorem, foreseen by Siméon Denis Poisson since 1813, and is concerned with complex analytical functions and integration. It is him which will define the concept of convergence uniform, thus sweeping the belief that the limit of a succession of continuous functions is always continuous. It makes in the same way with the series by defining the concept of absolute convergence and is prohibited, or almost, to summon the divergent series, contrary to its predecessors who write without formality

• In this context, Bernard Riemann resumes work of Tchebyscheff and in a report of 1859 in a decisive way research will make progress on the conjecture of Legendre. It uses for that, prolonging the methods of Tchebyscheff, the complex analysis, this still new theory in full effervescence.

• It extends initially the function $\backslash \; zeta$ of Euler to all positive realities larger than 1, then, reaches the complex values of the variable, than it calls $s=\; \backslash \; sigma+it$, with $\backslash \; sigma>1$. Lastly, using the properties of the function $\backslash \; Gamma$ d' Euler, it from of deduced a representation of $\backslash \; zeta\; (S)$ by a curvilinear integral, which then enables him to extend the function $\backslash \; zeta$ to the whole of the complex plan, except for $s=1$ whose Dirichlet had shown that it was about a simple pole of residue 1.

• It shows a fundamental relation called functional equation which connects the value of the function $\backslash \; zeta$ in $s$ to that in $1-s$

This relation shows that the axis $\backslash \; Re\; \{E\}\; (S)\; =1/2$ plays a fundamental role in the study of the function $\backslash \; zeta$. If the behavior of $\backslash \; zeta$ of this axis is known on the right, the functional equation makes it possible to supplement and one then knows all on $\backslash \; zeta$.

• Riemann shows easily that the function $\backslash \; zeta$ is not cancelled on the half-plane $\backslash \; Re\; \{E\}\; (S)\; >\; 1$, and thus, by the functional equation, $\backslash \; zeta$ is not cancelled either on , except the negative even entireties which one indicates by Zeros commonplace. It is in addition easy to show that each one of these zero commonplace is simple.

• Riemann, then shows that $\backslash \; zeta\; (S)$ cannot be cancelled, apart from the even negative entireties, that in the band $0\; \backslash \; the\; \backslash \; Re\; \{E\}\; (S)\; \backslash \; the\; 1$, and emits the following conjecture

• In the theory of the function $\backslash \; zeta$, in consequence of the theorem of factorization of Hadamard relating to the functions méromorphes of a nature finished $\backslash \; rho,$

For any function méromorphe $f\; (S)$ of a nature finished $\backslash \; rho$ there exist two entireties $m\_1$ and $m\_2$ smaller than $\backslash \; rho$, and a polynomial $q\; (S)$ of degree lower than $\backslash \; rho$ such as $f\; (S)\; =e^\; \{Q\; (S)\}\backslash \; frac\; \{p\_1\; (S)\}\{p\_2\; (S)\},$ where $p\_1\; (S)$ and $p\_2\; (S)$ are canonical products of functions of natures $m\_1$ and $m\_2$ built on the zero $a\_i$ and the poles $b\_i$ of $f$. $p\_1\; (S)\; =\; \backslash \; prod\_\; \{n=1\}\; ^\; \backslash \; infty\; \{E\; \backslash \; Big\; (\backslash \; frac\; \{S\}\; \{a\_n\},\; m\_1\; \backslash \; Big)\},$ $p\_2\; (S)\; =\; \backslash \; prod\_\; \{n=1\}\; ^\; \backslash \; infty\; \{E\; \backslash \; Big\; (\backslash \; frac\; \{S\}\; \{b\_n\},\; m\_2\; \backslash \; Big)\},$ with $E\; (U,\; m)\; =\; (1-u)\; e^\; \{u+u^2/2+\; \backslash \; ldots+u^m/m\},$

the zeros and the poles play a central role. For the function $\backslash \; zeta$ of Riemann, which is of order 1, one has

• One sees thus that the determination of the zero $\backslash \; rho$ is a key question. However these zero $\backslash \; rho$ are distributed symmetrically compared to the real axis since the function is real on the real axis (principle of symmetry of Schwarz), but répartissement also symmetrically compared to the axis $\backslash \; Re\; \{E\}\; (S)\; =1/2$.

• the simplest solution and most pleasant with the mathematician is that all the zeros noncommonplace $\backslash \; rho$ are on axis 1/2. One should not see another thing like initial motivation with the assumption of Riemann. This assumption is however heavy consequences. But neither Riemann nor its continuators will manage to show it.

• the remainder of the report establishes the link between the zeros of the function $\backslash \; zeta\; (S)$ and the functions of the arithmetic one, but the demonstrations are only outlined.

• First of all, it gives the number of zeros of the function $\backslash \; zeta\; (S)$ in the rectangle $\backslash \; times$ as being

It was made here use of the Notations of Pram (1909), which actually was used by Bachmann in its treaty Zahlentheorie Tome 2,1894 (page 402 for those which want to check) where $O\; (F)$ means that there exists a constant $A$ which raises the term represented by $Af\; (T)$ when $t$ is rather large.

• Then comes the bond between the function $\backslash \; pi\; (X)$ and the function $\backslash \; zeta\; (S)$ in the form

(with $\backslash \; Re\; \{E\}\; (S)\; >1$) that it is a question of reversing to obtain the theorem of the prime numbers. He writes for that

• to finish, Riemann claims that explains the conjecture of Legendre perfectly and that one can even deduce from it that $Li\; (X)$ raises $\backslash \; pi\; (X)$ with a term of error $O\; (x^\; \{1/2\})$. It is in fact a conjecture of Gauss that one has $\backslash \; pi\; (X)\; \backslash \; Li\; (X)$ (but it is false).

• This memory is the only report of Riemann concerning the theory of the numbers. Riemann dies in 1866, at the 40 years age.

Continuators of Riemann

• Since, the conjecture of Riemann, and the study of the function $\backslash \; zeta\; (S)$ occupies the spirit of many mathematicians, who do not measure all, far from there, the difficulty of the task bequeathed by Riemann. Because, let us say frankly, the demonstrations of Riemann are, when they exist, often incomplete, not to say frankly false. It will be necessary a long time to have a true demonstration of the theorem of the application conforms " of Riemann" for example. And that is also true for its report of 1859 on the function $\backslash \; zeta\; (S)$. However, the audacity of Riemann, which is a partisan convinced of the power of the methods of the complex variable, constitutes a revolution for the time.

• the formulas of the report of Riemann are shown by Hadamard in 1893 and Von Mangold in 1895 (with a small error concerning the formula on the number of zeros of $\backslash \; zeta$, repaired in 1905). Let us add that one showed that the assumption $\backslash \; pi\; (X)\; \backslash \; Li\; (X)$ involves the assumption of Riemann on the real part of the zeros of $\backslash \; zeta\; (S)$.

• One will show rather quickly the theorem

The three following proposals are equivalent:

• $\backslash \; pi\; (X)\; \backslash \; approx\; Li\; (X),$

• $\backslash \; pi\; (X)\; =\; \backslash \; sum\_\; \{N\; \backslash \; X\}\; \{\backslash \; frac\; \{\backslash \; Lambda\; (N)\}\{\backslash \; ln\; N\}\}\; \backslash \; approx\; Li\; (X),$
• $\backslash \; theta\; (X)\; =\; \backslash \; sum\_\; \{p\; \backslash \; X\}\; \{\backslash \; ln\; p\}\; \backslash \; approx\; x.$

Some examples of premature advertisements

• In 1883, Halphen (CRAS, 1883, T96, page 625 and following) announces

• Always in the reports of the Academy of Science, at dated July 13rd, 1885, one finds a note presented by Charles Hermite and written by Stieltjes, this one claims to have shown the assumption of Riemann on a small page!

The demonstration of Stieltjes is thus false.

• In a correspondence between Hermit and Stieltjes, Hermite requires of him to contact Mittag-Leffler which has just read its communication and which requires explanations. One is unaware of which was the answer of Stieltjes nor the content of the correspondence supposed between Stieltjes and Mittag-Leffler but it is perhaps there that it is necessary to find the unexplainable reason of the abandonment by Stieltjes of its research on the function $\backslash \; zeta$ and of which it wanted to make its thesis. It will indeed support a thesis written in three months, " Research on a few semi-convergentes" series; in 1885.

• In the same memory in which it shows the formula which bears its name, Jensen announces in 1899 in the acta mathematica which it will show the assumption of Riemann. No the continuation.

Majority

The great theorem of the prime numbers

• In 1896 Jacques Hadamard and Charles-Jean de la Vallee poussin shows independently the conjecture of Legendre in its final form:

Theorem of the prime numbers

by showing that the function $\backslash \; zeta\; (S)$ is not cancelled on a field $D=\; \backslash \; \{S\; \backslash \; in\; \backslash \; mathbb\; \{C\}\; |1>\; \backslash \; Re\; \{E\}\; (S)\; >\; 1\; \backslash \; frac\; \{has\}\; \{\backslash \; ln\; \backslash \; Im\; \{m\}\; (S)\}\backslash \}.$ where $A$ is an adequate constant.

• Until now, 2006, one did not succeed in much better doing: nobody succeeded in showing that $\backslash \; zeta\; (S)$ is nonnull on a tape $1$, some is $\backslash \; delta>0$, whereas the mathematical community in its very great general information believes that $\backslash \; zeta\; (S)$ is not cancelled on the tape $]\; 1/2,\; 1]$, in accordance with the assumption of Riemann.

• Edmund Pram, in a succession of memories and communications, will simplify the preceding evidence, will give a new demonstration of the postulate of Bertrand and will establish as of 1903 that the theorem of the prime numbers is equivalent to the assertion " The function $\backslash \; zeta\; (S)$ does not cancel on the line $\backslash \; Re\; \{E\}\; (S)\; =1$" , result establishes by Hadamard in 1892.

The remainder in the theorem of the prime numbers

Transcendent theorems and others

• It is with much difficulty that one had succeeded in showing the conjecture of Legendre-Gauss. Also, any theorem which was equivalent to the theorem of Hadamard-Of the Valley Chick was described as trancendant.

• In fact, one had noted that the initial transcendent theorems used in a major way the theory of the complex variable. As this conviction was forged as one could not do without the theory of the complex variable in the demonstration of the theorem of the prime numbers since this one was equivalent to show that the function $\backslash \; zeta\; (S)$ was not cancelled on axis 1 (theorem of Pram).

• the mathematicians thus distinguished these theorems from theory of the numbers which required the theory of the variable complexes by describing them as transcendent, while the other theorems had a demonstration described as elementary. This classification had the defect to be fluctuating. Thus, certain equalities had a member described as transcendent, while the other was not it! In addition, with time, one ended up finding demonstrations elementary for qualified theorems a time of transcendent.

• the death-blow was given in 1949 when Selberg gave finally an elementary proof of the great theorem of the prime numbers.

New tools

The difficulties encountered to justify the assertions of Riemann fully encourage the mathematicians to invent new tools. Two will be born in this orée of the twentieth century: the theory of the series of Dirichlet which will be started with Cahen, and the theory of the functions presques-periodicals, work of Esclangon, but mainly of Harald Bohr and Edmund Landau, and which will continue with Favard and Besicovitch before being absorptive by the harmonic analysis (Wiener,…).

Bohr and the theory of the almost periodic functions

• the theory of the functions presques-periodicals is primarily the work of Bohr and Landau as from 1909, even if Esclangon proposed with the whole beginning of the twentieth century a close theory, that of the functions quasi-periodicals. The objective is the generalization of the Fourier series and the study of the properties of these functions. In the Fourier series of a function, the coefficients, one remembers it, are calculated starting from an integral of the function, presumedly periodic. The function is written then as a sum of goniometrical functions whose frequency is a multiple of the period of the function. And, without much difficulty, one passes from the traditional representation by sine and cosine to a representation utilizing exponential whose argument is imaginary pure.

• In the theory of the functions almost-periodicals, is one gives itself the coefficients " of Fourier" of a trigonometrical series and one studies the properties of them, is one seeks which must be the properties of a function so that it is " presque" periodical. It is shown that the two points of view coincide if one remains reasonable in his requests.

• One says that a function $f$, definite and continues on $\backslash \; mathbb\; \{R\}$, is almost-periodical (within the meaning of Bohr) if there exists a $L>0$ number such as for any interval length $L$ and for all $\backslash \; epsilon>0$ exists a number $\backslash \; tau=\; \backslash \; tau\; (\backslash \; epsilon)\; \backslash \; Leq\; L$, called $\backslash \; epsilon$-presque period, such as

• This definition extends to the complex functions and it is said that $f$, function analytical complexes, is almost-periodical in the band $$ if $f\; (\backslash \; sigma+it)$ is almost periodic in $t$ for $\backslash \; sigma\; \backslash \; in$.

• the theory of the periodic presques functions shows that any function almost-periodical is limited and that an analytical function complexes can be almost-periodical in a band only if it remains limited there.

• the sum and the product of functions almost-periodicals are almost periodic.

• One shows then that a function almost-periodical is the uniform limit of a succession of functions almost-periodicals, that the derivative of a function almost-periodical which is uniformly continuous is itself almost-periodical…

• the first result of the theory is that any almost periodic function admits a median value

result which one deduces the second result from the theory, and who relates to the representation in generalized Fourier series

Any function $f$ almost-periodical is written

• Then one shows that

and there is an inequality of the kind Inégalité of Bessel:

There exists a $A>0$ number and as large as $t\; >\; t\_0$ such as $is$ t\_0$|\backslash \; zeta\; (1+it)|>\; \backslash \; ln\; \backslash \; ln\; T\; has.$

As $1/\backslash \; zeta\; (S)$ exists for all $s$ of higher real part or equalizes to 1, and that she does not admit any pole on axis 1, one from of deduced that she is not limited on this axis.

• In addition, because of the almost-periodicity on the half-plane $\backslash \; sigma>1$, with all $\backslash \; epsilon>0$ there exists an infinity of values of $t$

The general theory of the series of Dirichlet

• the theory general of the series of Dirichlet is started with Emile Cahen in his thesis On the function of Riemann and similar functions constant on March 16th, 1894. This work is the object of very serious reserves on behalf of the mathematicians of this time but will be used of framework and guide for the following studies because it tries to make a systematic theory of the representable functions by series of Dirichlet.

• the theory is articulated on the concept of X-coordinate of convergence as soon as is shown that the convergence of the series for a value $s\_0=\; \backslash \; sigma\_0+it\_0$ involves convergence for the values $s=\; \backslash \; sigma+it$ with $\backslash \; sigma\; >\; \backslash \; sigma\_0$ (theorem of Jensen, 1884). And several X-coordinates of convergence are classically now defined. It there with the X-coordinate of absolute convergence which corresponds to the X-coordinate of convergence of the series of Dirichlet whose coefficients are the absolute values of the coefficients of the starting series. This X-coordinate will be noted $\backslash \; sigma\_a$. The initial series admits it a X-coordinate of convergence known as simple noted $\backslash \; sigma\_s$. $\backslash \; sigma\_s\; \backslash \; the\; \backslash \; sigma\_a$.

• In the simple half-plane of convergence, the sum of the series of Dirichlet represents an analytical function complexes regular, and its derivative is itself analytical regular in the same half-plane.

• Theorem (Cahen, 1894)

Is a series of Dirichlet $f\; (S)\; =\; \backslash \; sum\_\; \{n=1\}\; ^\; \backslash \; infty\; \{a\_n\; e^\; \{-\; \backslash \; lambda\_n\; S\}\}$ which one supposes that the simple X-coordinate of convergence is positive or null. Then the X-coordinate of convergence is given by $\backslash \; sigma\_s=\; \backslash \; limsup\_\; \{N\; \backslash \; rightarrow\; \backslash \; infty\}\; \{\backslash \; frac\; \{\backslash \; ln\; |With\; (N)|\}\; \{\backslash \; lambda\_n\}\}$ where $A\; (N)\; =\; \backslash \; sum\_\; \{K\; \backslash \; N\}\; \{a\_k\}$ .

Let us make two small applications to the function $\backslash \; zeta$ of Riemann.

There is for $\backslash \; zeta\; (S)$ $a\_n=1$, and $\backslash \; lambda\_n=\; \backslash \; ln\; n$. From where $|With\; (N)|=n$ and the formula gives $\backslash \; sigma\_s=1$.

On the other hand for the function $\backslash \; frac\; \{1\}\; \{\backslash \; zeta\; (S)\}=\; \backslash \; sum\_\; \{n=1\}\; ^\; \backslash \; infty\; \{\backslash \; frac\; \{\backslash \; driven\; (N)\}\{n^s\}\},$ one has $a\_n=\; \backslash \; driven\; (N)$, $\backslash \; lambda\_n=\; \backslash \; ln\; n$ and $A\; (N)\; =M\; (N)$ function sommatoire of the function of Möbius. Let us suppose $M\; (N)\; =\; O\; (n^\; \backslash \; theta)$. The formula then gives for X-coordinate of convergence the value $\backslash \; theta$. And the function will be regular for $\backslash \; sigma\; >\; \backslash \; theta$ thus $\backslash \; zeta\; (S)$ will not be cancelled on the half-plane $\backslash \; sigma\; >\; \backslash \; theta$.

One sees thus that there exists a bond between the function $\backslash \; zeta\; (S)$, the assumption of Riemann, and the function sommatoire $M\; (X)$. In fact, thanks to the Formula sommatoire of Abel, one with the integral formula $\backslash \; frac\; \{1\}\; \{\backslash \; zeta\; (S)\}=s\; \backslash \; int\_1^\; \backslash \; infty\; \{\backslash \; frac\; \{M\; (U)\}\{u^\; \{1+s\}\}\; of\; the\},$ which shows that any assumption of growth on $M\; (X)$ is translated immediately on the convergence of the integral. Such assumptions were formulated at various times and bear the name credits of assumptions of Mertens.

• the theory of the series of Dirichlet seeks then the order of the function $f\; (S)$ represented by the series. Thus the concept of a finished nature is defined (distinct from that implied by the theorem of Phragmen-Lindelöf). It is shown indeed that $f\; (S)$ is $o\; (T)$ in a half-plane vaster than its half-plane of convergence. Also, one defines $\backslash \; driven\; (\backslash \; sigma)$ smallest of the $\backslash \; xi$ such as $f\; (\backslash \; sigma+it)\; =O\; (t^\; \backslash \; xi)$. The number $\backslash \; driven\; (\backslash \; sigma)$ thus defined is called the order finished of $f\; (S)$ on the line $\backslash \; sigma$.

Applying the theorem of Lindelöf

Is $f\; (\backslash \; sigma+it)$ an analytical function complexes which is O (exp (E T)) in a band $\backslash \; sigma\_2$ for all e>0; If it is $O\; (t^a)$ on $\backslash \; sigma=\; \backslash \; sigma\_1$ and $O\; (t^b)$ on $\backslash \; sigma=\; \backslash \; sigma\_2$

then F is $O\; (t^\; \{K\; (\backslash \; sigma)\})$ in the bande$$ where $k\; (\backslash \; sigma)$ is the function $k\; (\backslash \; sigma)\; =\; \backslash \; frac\; \{(\backslash \; sigma\; \backslash \; sigma\_1)\; b+\; (\backslash \; sigma\_2-\; \backslash \; sigma)\; has\}\; \{\backslash \; sigma\_2-\; \backslash \; sigma\_1\}$

Bohr deduced the following theorem from it: the function $\backslash \; driven\; (\backslash \; sigma)$ is a convex, positive and decreasing Fonction of $\backslash \; sigma$.

For the function $\backslash \; zeta\; (S)$, we know that $\backslash \; driven\; (\backslash \; sigma)\; =0$ if $\backslash \; sigma>1$ since it is limited on the half-plane of convergence. In addition it is shown that $\backslash \; driven\; (\backslash \; sigma)\; =1/2-\; \backslash \; sigma$ if $\backslash \; sigma\; \backslash \; the\; 0$ by the functional relation. It is thus a question of connecting the item (0,1/2) to the item (1,0) by a decreasing and convex positive curve. The line which joint these two points is of equation $1/2-\; \backslash \; sigma/2$ and that give $\backslash \; driven\; (1/2)\; \backslash \; the\; 1/4$. One showed up to now $\backslash \; driven\; (1/2)\; \backslash \; the\; 139/858$ (Kolesnik). One conjectures that $\backslash \; driven\; (1/2)\; =0$ (assumption of Lindelöf).

Conjectures of Mertens, Von Sterneck…

It is time maintaining to speak about the great conjectures which are subjacent with all that has just been known as.

Various conjectures of Mertens

The first conjecture which solves the assumption of Riemann is that of Mertens. One remembers that Stieltjes in his " preuve" from 1885 had affirmed that it was easy to see that the series of Dirichlet which defined $1/\backslash \; zeta\; (S)$ converged provided that $\backslash \; Re\; \{E\}\; (S)\; >1/2$. This assertion was equivalent to affirm that $M\; (U)\; =O\; (u^\; \{1/2+\; \backslash \; epsilon\})$, some is $\backslash \; epsilon$.

In an article of 1897, using a numerical table up to 10.000, Mertens notes that $|M\; (N)|\; \backslash \; the\; \backslash \; sqrt\; \{N\}$ for , result confirmed soon by Von Sternek, in 1901, up to 500.000, and with the international congress of the mathematicians of 1912 per 16 values in lower part of 5 million. This one benefits from it to propose increase $|M\; (N)|\; \backslash \; the\; \backslash \; sqrt\; \{N\}\; /2$ which will be refuted in 1963 by Neubauer numerically by showing that $M\; (7\; 760.000.000)\; =47465$, which is higher on the terminal of Von Sterneck. Let us specify that Jurkat, in 1973, will show that the conjecture of Von Sterneck is asymptotically false.

The conjecture of Mertens arises in three forms

• the normal form $|M\; (N)|\; \backslash \; the\; \backslash \; sqrt\; \{N\}$,

• the generalized form, there exists $A>0$ such as $|M\; (N)|\; \backslash \; has\; \backslash \; sqrt\; \{N\}$,
• the weakened form $\backslash \; int\_1^x\; \{\backslash \; frac\; \{M^2\; (U)\}\{u^2\}\; of\; the\}\; =O\; (\backslash \; ln\; X).$

The weakened form implies the generalized form which implies the normal form.

Current results on the conjectures of Mertens

• Currently, one knows that the first form is false (Odlyzko and Te Riele, 1985) but the proof given does not make it possible to answer on the two other forms. One currently conjectures that the second form is also false. However, the formula sommatoire of Abel applied to $1/\backslash \; zeta\; (S)$ watch which one has, if the assumption of Riemann is true, $M\; (U)\; =O\; (u^\; \{1/2+\; \backslash \; epsilon\}).$

• As one saw in connection with the series of Dirichlet, the assumption of Mertens, whatever its form, implies the assumption of Riemann, but it has an interesting consequence, the simplicity of the zeros of the function $\backslash \; zeta$. However precisely one calculated million zeros of the function $\backslash \; zeta$ and one all found them of real part equalizes with 1/2 and simple.

The assumption of Lindelöf

• One tried to show weaker versions of the assumption of Riemann as time passed and that no progress was obtained in this way. Thus, the behavior of the function $\backslash \; driven\; (\backslash \; sigma)$ which is decreasing and convex, combined with the assumption of Riemann leads Lindelöf to conjecture that $\backslash \; driven\; (1/2)\; =0$ i.e. one has

• In its thesis, constant on June 20th, 1914 On the whole functions of zero value and a finished nature and in particular the functions with regular correspondence , Georges Valiron states in an application a theorem which proves to be essential in the theory of the function $\backslash \; zeta$.

There exists a number $\backslash \; delta>0$ such as in any interval $T+1$ exists an infinity of values of $t$ for which $\backslash \; zeta\; (\backslash \; sigma+it)\; \backslash \; Ge\; t^\; \{-\; \backslash \; delta\},$ and that some is $\backslash \; sigma\; \backslash \; in$. The importance of this theorem is due to the fact that it is the only one which allows a crossing of the band criticizes $$.

• the theory of the series of Dirichlet shows that $\backslash \; delta\; \backslash \; the\; 1$ without assumption but one does not know a value of $\backslash \; delta$ without additional assumption. On the other hand, if one admits the assumption of Lindelöf, then $\backslash \; delta$ can be taken as small as one wants, and that is also worth for the assumption of Riemann. Indeed, it is shown whereas one has a theorem of Valiron with for $\backslash \; delta$ a decreasing function tending towards 0 as $t$ tends towards the infinite one.

• the theorem of Valiron is primarily used for raising (or undervaluing) complex integrals utilizing the function $\backslash \; zeta\; (S)$ on a way crossing the critical band.

• It is thanks to him which one shows that the weakened assumption of Mertens (and thus other assumptions) imply not only the assumption of Riemann but also the simplicity of the zeros of the function of Riemann and certain decreases between the zeros of the function $\backslash \; zeta$.

The theorem of Hardy

• In a communication with the Academy of Science of Paris, in 1914, Godfrey Harold Hardy shows that the function zeta cancels an infinity of time on axis 1/2. This beautiful result will be moderated thereafter by the comparison between the proportion of the zeros on the axis in lower part of T with those awaited: it is very weak.

The theorem of Speiser

• In a communication made in the Reports of the Academy of Science of Paris in 1912, Littlewood announces to have shown " of a theorem of Misters Bohr and Landau" the following theorem

Or the function $\backslash \; zeta\; (S)$, or the function $\backslash \; zeta\text{'}\; (S)$ has an infinity of zeros in the half-plane $\backslash \; sigma>1-\; \backslash \; delta$, $\backslash \; delta$ being a positive quantity arbitrarily petite.

• It seems that Littlewood did not publish its result.

• the question of the number of zeros of $\backslash \; zeta\text{'}\; (S)$ then takes a new turning in the theorem of Speiser, published in the mathematische annalen, year 1935. This one, analyzing the surface of Riemann from the point of view of the lines of constant argument, and using the functional relation fully, arrives at the conclusion

The assumption of Riemann is equivalent to the absence of zero noncommonplace of derived the $\backslash \; zeta\text{'}\; (S)$ in the half-plane .

Theorems of oscillation

• From time immemorial one noted that it seemed to exist a rather sensitive prédominence of the prime numbers of the form 4n+3 compared to those of the form 4n+1.

It is with the demonstration of this result that will harness Phragmen.

• First of all it obtains the following theorem

Theorem of Phragmen (1891):

That is to say F (U) a real function locally integrable for U >1. It is supposed that the integral

Then

1/ some is $\backslash \; delta>0$, it does not exist any $x\_0$ such as for all $x\; >\; x\_0$, one has $f\; (X)\; >\; \backslash \; delta$ (respectively ),

2/ so moreover F (X) has an infinity of points of discontinuities, the divergence of a certain series formed starting from discontinuities and of the jumps implies that F (X) oscillates indefinitely around 0 (infinity of change of signs).

who is historically the first theorem of oscillation and of which it is used for to show

• calculation by Jørgen Pedersen Gram of noncommonplace first zeros of $\backslash \; zeta\; (S)$ in 1895 and 1903, makes it possible Erhard Schmidt to show in 1903 the first theorem of oscillation which is not an alternation of signs:

Theorem of Pram (1905):

That is to say F (U) a real function locally integrable. The function is defined

If there exists a $u\_0$ such as $f\; (U)\; >0$ for all $u\; >\; u\_0$, then $\backslash \; sigma\_c$ is a singularity of $\backslash \; phi\; (S)\; \backslash ,$.

from which one deduces that if $\backslash \; sigma\_c$ is not a singularity, then F does not keep a constant sign.

• an immediate application of the theorem of Pram with $1/s\; \backslash \; zeta\; (S)$ for which the function F (U) of the theorem is worth M (U), function sommatoire of Möbius, gives, since $1/s\; \backslash \; zeta\; (S)$ is regular on the real interval

How much zeros on axis 1/2

• Hardy had shown in 1914 qu ' there was an infinity of zeros on axis 1/2. But its demonstration gave only one negligible quantity compared to the number of the zeros in the critical band. One thus sought to supplement the theorem of Hardy.

• One classically calls No (T) the number of the zeros on axis 1/2 and of positive imaginary part lower than T and NR (T) the number of zero the noncommonplace ones in the band criticizes and of positive imaginary part lower than T. the assumption of Riemann states than No (T) =N (T). But one is unaware of still today if No (T) ~N (T). The question of the comparison of No (T) and NR (T) arises as of the demonstration of the theorem of Hardy. Let us recall that NR (T) believes like T ln (T).

• Initially, in 1921, Hardy and Littlewood show that there exists constant has for which one has No (T) > AT.

• Selberg, in 1942, taking again the demonstration of Hardy and Littlewood succeeds in improving the estimate to No (T) > HAS T ln (T). From now on there is the good order and work will concentrate with the improvement of the report/ratio No (T) /N (T).

• In 1974, Levinson showed that the report/ratio is at least equal to 1/3 when T tends towards the infinite one.

• Contrary, a theorem of Bohr and Landau of 1914 shows that the growth of $|\backslash \; zeta\; (S)|^2$ is related to the distribution of the zeros. The median value of $|\backslash \; zeta\; (S)|^2$ is raised on the lines $\backslash \; sigma=cte$. One from of deduced that the proportion of the zeros apart from the band tends towards 0 when T tends towards the infinite one. That carries out straight to the assumption of density.

The theorem of universality of Voronin

There for a long time exist theorems known as of universality which express that a given function approach any function analytical in a given surface. Such theorems were shown between the two world wars by various authors. Voronin showed in 197x that the function zéta of Riemann had this property. Thereafter Bagchi extended this result and one calls thus theorem of Voronin-Bagchi the following statement:

That is to say $\backslash \; epsilon\; >0$ fixed. For very compact K included in the band] 1/2; 1 and for any function F analytical not cancelling itself on K, there exists a $t\_0$ such as for all $s\; \backslash \; in\; K$ one has $|\backslash \; zeta\; (s+it\_0)\; -\; F\; (S)|\; \backslash \; the\; \backslash \; epsilon.$ {{fine quotation}}

The function sommatoire of Möbius

• the function $M\; (X)\; =\; \backslash \; sum\_\; \{N\; \backslash \; X\}\; \{\backslash \; driven\; (N)\}$ is called function sommatoire (the function of) Möbius. One knows only little thing on this function without more or less strong assumption. Apart from the conjectures of Mertens which are the subject of a special paragraph, one knows that $M\; (X)\; =o\; (X)$, this statement being equivalent to the theorem of the prime numbers, and one succeeded in showing that $M\; (X)\; =O\; (X\; (\backslash \; ln\; X)\; ^\; \backslash \; alpha)$ for all $\backslash \; alpha>0$.

Under the assumption of Riemann, one can show that one has $M\; (X)\; =O\; (x^\; \{1/2+e\; (X)\})$ where the function E (X) tends towards 0 when X tends towards the infinite one. Titchmarsh showed that E (X) =O (1/ln ln X).

• the theory of the function $M\; (U)$ is very obscure, even with strong assumptions. The best currently known result is a light improvement of an already known result of Pram in 1909:

One succeeded in improving only the power of the $\backslash \; ln\; u$ which passed from $1/2$ to $3/5$ in almost a century!

• If there exists one zero of the function of Riemann in $s=\; \backslash \; beta+i\; \backslash \; gamma$, one shows that $M\; (X)\; =O\; (x^\; \{\backslash \; beta+\; \backslash \; epsilon\})$ for all $\backslash \; epsilon>0$.

• the assumption of Mertens weakened

has very interesting consequences:

• It implies the assumption of Riemann.

• the zeros of the function of Riemann are simple.

Method of Vinogradov

• Vinogradov, continuing the searchs for Hardy, Littlewood and Weyl on the estimate of the trigonometrical sums finally manages (after several communications dating from the years 1930) to show that one has