Ilan Vardi
IHES, Bures-sur-Yvette
14 décember 1998
Résumé de Cyril Banderier et Ilan Vardi
``Le plus court chemin entre deux vérités dans le domaine réel passe par le domaine complexe.'' (Jacques Hadamard)
Cette citation illustre la puissance que l'analyse peut apporter
lorsque l'on est confronté à des questions de théorie des nombres.
Le plus ancient et le plus fondamontal de ses thèmes est celui
des nombres premiers. La première question qui se pose est :
"Y a-t-il une infinité de nombres premiers ?"
Il y a de multiples preuves élémentaires :
(1) |
Cette preuve peut être modifée en remarquant que , où . S'il y avait un nombre fini de nombres premiers alors (1) impliquerait que serait rationnel, or Legendre a prouvé que ce n'était pas le cas en 1797, voyez également [6]. Diverses autres preuves sont données dans [7].
Une version améliorée de cet argument est due à Mertens: la version finie de (1) donne
en prenant les logarithmes, on obtient(2) |
Laquelle de ces preuves est la "meilleure" ? Un point de vue sage serait de choisir celle qui offre la meilleure généralisation. Par exemple, la preuve d'Euclide montre facilement qu'il y a une infinité de nombres premiers de la form 4k+3 (considérez 4 p1 p2 ... pn-3, mais il faut légèrement adapter l'argument pour prouver l'infinitude l'infinitude des nombres premiers de la forme 4k+1 (il faut considérer cette fois 4 (p1 p2 ... pn)2+1). Plus généralement, on aimerait démontrer l'assertion de Dirichlet (ce qu'il a effectivement fait en 1837, dans [3]) ``il existe une infinité de nombres premiers de la forme ak+b, où a et b sont premiers entre eux.'' Il s'avère que la preuve de fait profond utilise une généralisation de la méthode d'Euler, i.e. l´équation (2):
Soit un caractère multiplicatif modulo q, c'est-à-dire une fonction à valeurs complexes vérifiant and (ce qui implique que si alors c'est une racine de l'unité et a donc norme 1). Un exemple est le symbole de Legendre (ou de Jacobi si q n'est pas premier)
En fait, il y a exactement caractères multiplicatif modulo q, tous donnés par où est une racine primitive et est tel que . L'importance des caractères se présent au regard des relations d'orthogonalité:(3) |
(4) |
This is definitely true for complex characters since otherwise, would imply that and since these terms are different, this would imply that , which is false as taking logs gives
and so the value at s=1 must be positive, hence the last sum in relation (4) is bounded.The real problem is then to bound the middle sum in relation (4), that is to say to show that . Dirichlet proved this result by a very ingenious method: He evaluated this number in closed form! This is now known as Dirichlet's class number formula:
where h is the class number of and its fundamental unit and w the number of roots of unity in this field (see the canonical reference [2]). Since each of these quantities counts something, so they are positive, the result now follows:Simpler proofs using only complex analysis are also possible. The idea is to use Landau's theorem that a Dirichlet series with positive terms has a pole at its abscissa of convergence and apply it to which has just been shown to have positive coefficients.
The distribution of primes is quite irregular, so it is easier to study their statistical behaviour. In this direction, let be the number of primes . Gauss conjectured that This assertion simply says: ``the probability that n is prime is about .'' This result was finally proved by Hadamard and de la Vallée Poussin in 1896. Both of them used fundamental ideas of Riemann who was the first to introduce complex analysis in the study of the distribution of prime numbers.
Using Perron's formula, namely
and using residues, Riemann essentially found what is perhaps the most important formula in analytic number theory (the von Mangoldt explicit formula):(5) |
Following Chebyshev, one defines and ,where when n=pm, and zero otherwise. A fairly straightforward partial summation shows that the prime number theorem is equivalent to (note that trivially, ), and that more generally,
One can then see from the explicit formula (5) that the prime number theorem would follow if one can bound , since each error term would then be of order < x. The prime number theorem would then be equivalent to showing that for . In fact, this is an equivalence (as was later shown by Wiener) and Hadamard and de la Vallée Poussin were able to prove that using some ingenious trigonometric identities. We will give a proof due to Mertens, in 1898. Set , then when (we restrict to ). But, by the Euler identity, one has and so Mertens' trick consists in noticing that , thus , hence .All numerical evidence shows that and it was long believed that this would be true for all x. Similarly, Chebyshev noted that the number of primes of the form 4k+3 seemed to be more abundant than the primes of the form 4k+1, more precisely, let then .
In fact, Littlewood proved in 1914 that changes sign infinitely often and the same is true for . In 1957 Leech showed that is first true for x = 26861, and that the similar inequality is first true for x = 608981813029 was shown by Bays and Hudson in 1978. No example of is known. Skewes first gave an upper bound which was later reduced by Sherman-Lehman and then te Riele [10] who gave an upper bound of 10370.
This behaviour can easily be explained using explicit formulas. In the case of , the point is the following: The explicit formula (5) expresses as a sum of powers . Assuming the Riemann Hypothesis, one can write this as
One can now see the reason for the bias: The function does not count primes but prime powers so what one really wants is the behaviour of which is given by so that The function is a very slowly oscillating trigonometric series which should be zero on average, so the extra term biases to be smaller than x on average. A simple description is that counts the number of prime powers , so the number of primes should be slightly less since the number of prime squares is of the same order as the error term.There is a similar explanation for the bias in arithmetic progressions. There is an explicit formula
where the Generalised Riemann Hypothesis has been assumed (there is no x term since is no longer a pole if ). As before one has but one really wants to look at y2= a (mod q). In particular, the same argument shows that there will always be fewer primes in the progression qn + a when a is a residue than when a is a nonresidue. Simply put, the ``balanced'' count is the set of prime powers = a (mod q) when a is quadratic residue since the number of prime squares congruent to a is of the same order as the error term in the analytic formulas.In 1994, Rubinstein and Sarnak (see [8]) were able to make Chebyshev's bias precise. Assuming GRH (if this is false, then there is no bias) and also the Grand Simplicity Hypothesis (GSH: All the ordinates of zeroes of L-function are linearly independent over ), then