On the "statistics" of primes

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On the ”statistics” of primes

B. Julia

To cite this version:

B. Julia. On the ”statistics” of primes. Journal de Physique, 1989, 50 (12), pp.1371-1375.

�10.1051/jphys:0198900500120137100�. �jpa-00211002�

Short Communication

On the "statistics" of primes

B. Julia

Laboratoire de physique théorique ^de l’Ecole Normale Supérieure, ^{24 rue} Lhomond, 75005 Paris France

(Reçu le 31 Mars 1989, accepté le 20 avril 1989)

Résumé.

On peut réinterpréter les séries de Dirichlet de la théorie multiplicative des nombres

comme des fonctions de partitions ^de systèmes quantiques solubles. En particulier ^le pôle de la fonc- tion zèta de Riemann à s

1 peut-être compris ^{comme une} "transition de phase". ^Plus précisement

la densité exponentielle des niveaux provoque ^une "catastrophe ultraviolette de Hagedorn". ^{Nous il-}

lustrons le dictionnaire entre la théorie des nombres et des cas particuliers ^{de la} mécanique statistique quantique ^{en étudiant} quelques exemples intéressants de gaz ^de particules libres, l’un d’eux réalise

une équivalence ^{entre bosons et} fermions. On connaissait le lien entre transitions de phase ^{et zéros}

de la fonction de partitions, ^{on trouve ici un} pôle explicite.

Abstract.

The Dirichlet series of the multiplicative theory of numbers can be reinterpreted

partition functions of solvable quantum mechanical systems. ^In particular ^the pole of the Riemann zeta function at

1 can be understood as a "phase transition". More precisely ^{there is an ultra-}

violet "Hagedorn catastrophy" ^{due to the} exponential density ^{of the} energy ^levels. We illustrate the

dictionary between Number Theory ^and special ^{cases of Quantum} Statistical Mechanics by studying

number of interesting gases ^{of free} particles, ^{in one} example ^{we obtain a} boson-fermion equivalence.

Phase transitions are usually associated with zeroes of the partition ^{functions here we have}

explicit pole.

0210 - 05.30 - 05.70F - 64.60C

This paper ^{is a short} announcement of some results on the generating series of the so called

"multiplicative" arithmetic functions. A more detailed treatment will be submitted to the Compte-

Rendus of the French Academy of Sciences (Math section).

The theory of Dirichlet series : D (s) = M ^An exp (-sBn) , splits ^{into two branches.}

n =

1...

Firstly there’ is the familiar case with Bn

namely ^the power series. These series have long

been recognized ^as statistical mechanical partition functions; strictly speaking ^{this is so in non}

relativistic cases. Thé varaible s is the inverse temperature, in natural units it is the inverse of an energy. Typically ^a "canonical ensemble" partition function is an expression of the form

with énergies En and statistical factors or degeneracies ^{S’n .}

Let us recall that the inverse of the discriminant of Dedekind :

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198900500120137100

1372

where _q

exp (2xiz) ^{is the} generating function for the numbers of internal "transverse"

(26 - ²

24) ^{modes of} equal "energy" ^{for the vibration of the} open ^bosonic relativistic string

in 26 dimensions (Veneziano-Nambu-Goto model). ^This energy ^{is to be understod as a} lightlike component ^of energy-momentum.

Let us take a second example : ^{the theta} function of an even integral lattice ^P [ibidem] :

generates ^the dimensions of the spaces ^{of states of} equal ^{mass for a} particle moving ^{on the dual} (in ^{the sense} of Fourier transform) ^{torus to the} lattice. I /this ^{particle is thought of as the center of}

mass of a bosonic string ^the product of the above two "partition" ^functions describes the full theory

of strings ^{in the} light ^cone gauge. (It is also the gener ing function of the basic representation ^of

a corresponding ^affine Kac-Moody algebra ^{when the torus is a} Weyl ^{torus for a} finite dimensional Lie group.)

These power ^series belong ^{to the} "additive" theory of numbers : our two examples ^describe roughly speaking respectively the number of partitions ^of integers ^{into sums of d} integers ^{and into}

sums of squares ^{of d} integers. The index n is proportional ^{to an} energy ^{in the} physical ^{models and,}

as such, it behaves necessarily additively ^{when one} considers the juxtaposition (

^{the tensor} prod- uct) of several of them. In particular ^{the sums above} correspond ^{to the} independent contributions of d effective dimensions.

Let us now consider the second branch of the theory of Dirichlet series : the case Bn

Log

If there is a physical interpretation ^{of such a} Dirichlet series as a partition ^function, the inverse

temperature in natural units should be

again ^{and the} energies ^the Bn’s. Then under exponentia-

tion the additivity ^of energy (in the absence of interactions) should become the multiplicativity ^of

the index n (for ^the product ^{of two} partition functions). Therefore it is not a surprise that indeed the "multiplicative" theory of numbers has a nice interpretation ^{in terms of free gases of various}

kinds of particles.

We may begin with the familiar Riemann zeta function : An

^{1. It is a famous} hypothesis ^of

Riemann [2] ^that the nontrivial zeroes of this function lie on its "symmetry" ^axis namely ^{Re s}

1 /2 .It ^{has a} pole ^{at s}

¹ with residue 1. Let us note that in 1859 Boltzmann was 14 years ^old.

In general ^a Dirichlet series converges ^{in a} right half-plane ^{Re s} > a,

is called the abscissa of convergence. The Riemann hypothesis ^is equivalent ^to saying that the function 1/zeta ^has 1/2 ^for

its abscissa of convergence. Despite ^{a lot of} progress ^{in our} mathematical understanding ^{of this}

function and of its many generalizations, the abscissa of convergence ^of 1/zeta has refused to go left of s

1. There is certainly ^a nasty phenomenon ^{at that} "temperature". ^In physical ^terms

the convergence ^{we are} talking about is the convergence ^{of low} temperature series with expansion parameter exp ( -1 / T) .

As we guessed ^{above the} natural energy ^{is the} logarithm ^{of the} integer

^{Let us} apply ^{the tech-} nique ^of particle physics and break the number n into elementary consistuents : namely primes.

The unique factorization of an integer ^into primes translates into a unique decomposition ^of Log n= L NpLog ^{p where the} Np’s ^{are also} arbitrary integers. ^The following ^{lemma is now clear :}

The zeta function is the grand ^canonical partition function of a "Riemann gas" ^of quantum

bosons of energies Log ^{p for all} primes ^{p at zero chemical} potential ^{and at inverse} temperature

s ⁼

1/T.

ity). Second the Boltzmann weight is that of grand ^canonical partition function because generic integers ^{are divisible} by arbitrary powers ^of primes, ^so the number of factors of the prime ^decom- position ^is oqual ^to the number of particles and takes all integer ^{values. But,} thirdly ^{the chemical} potential ^{is set} equal ^{to zero or} equivalently fugacity ^{is set} equal ^{to one.} Physicists naturally ^would

consider the more general ^{case of an} arbitrary ^chemical potential because for a finite dimensional vessel and unless there is Bose condensation the chemical potential ^is negative ^{at thermal} equilib-

rium.

So let us restore the fugacity f

exp (8 J-l) the Riemann zeta function is generalized by ^the

new Dirichlet series with coeficients A"

¡1I(n) ^where

(n) is the number of terms appearing ⁱⁿ

the decomposition ^{of n into} prime ^factors (counting multiplicities).

The special ^case f ^{== -1 is} easily computable ^and gives ^{the ratio zeta} (2s) /zeta (s) , this is the

case of a purely imaginary ^chemical potential ^The convergence abscissa if we assume the Riemann

hypothesis ^{is Re s} > 1/2, ^{but from the} point of view of thermodynamics ^{we are} interested in the

logarithm ^{of the} partition function : a zero at s

1 is just ^{as bad as a} pole although ^its physical interpretation ^may be different

Let us pause ^{for a} minute and compute ^the density ^{of states} of the Riemann gas. ^{The levels}

are En

Log n ^{and their} spacing ^is asymptotically Log (1 ⁺ n) - Log n

1 / n (1 - 1/2n ⁺ ...) ^so

their density ^is p (E)

exp (E) ⁺ 1/2 ^{+ 0} (exp (-E)) . ^{Such an} exponential degeneracy ^{has been}

encountered before for example ⁱⁿ string theory (up ^{to an} important power) ^{or in other} systems

with an infinite number of degrees ^{of freedom.} Using ^the Weyl formula for the asymptotics ^of

the density ^{of states of a free} particle ^{in a} piecewise smooth and bounded box the Riemann gas could live only ^{in an} infinite dimensional space ^{or at least in an} exotic finite dimensional one...

The increase in the density ^{of states is} responsible ^{for a} catastrophy ^at the inverse temperature

s ⁼

1 below which one does not know how to compute ^the partition ^{function one} spoke ^{of fire-}

balls see Hagedorn [3]. ^The understanding ^{of that} phenomenon ^would help proving ^{the Riemann} hypothesis. ^{Let us} emphasize that there is a lot of research going ^{on in the} community ^of strong interaction theorists right ^{now on this} problem in relation with the deconfinement of quarks ^{in hot}

and dense matter. We shall return to this problem ^{in a further} publication.

Let us finally give ^{a few more} examples of statistical mechanical ensembles of number theory.

Our first hint of the above scheme came from the so-called Môbius transform of number theory.

Given a first Dirichlet series D (s) with coefficients An ^we may ^tensor the system ^{with a Riemann} gas ^and look for the coefficients of the product ^zeta (s) ^D (s) . They ^{are the} A’ , = M Ad for ^all

d 1 n

divisors d of

The nice result of Môbius is that the transformation can be inverted and he found that An == E Ai p (n/d) where the Môbius function M (n) vanishes if n is divisible by ^a square and is equal ^{to -1 to} the number of factors of n otherwise (the latter numbers are the so-called

squarefree ^or quadratfrei ^numbers, particle physicists would call them supernumbers ^{or s- numbers}

as we shall see).

What is the gas corresponding ^{to the} partition ^function 1/zeta(s) ? ^{It would} provide ^{a re-}

alization of the inverse Môbius transform as a tensor product ^{with this gas.} The coefficients of this Dirichlet series are precisely the p (n)’s. ^{The answer} is that it represents ^a gas ^{of free} ghost

fermions with the same energies Log ^p. They ^are fermions because one cannot put ^{two of them in}

the same state and ghosts because of the minus sign ^{in the norm} (or ^the probability) ^{of all states}

with an odd overall number of fermions.

Actually ^{one can} also consider a gas of bona fide free fermions. The corresponding weights

are An == IJ-l (n) 1 ^{. It is} again easy ^to compute ^{the sum : it is} equal ^{to zeta} (s) /zeta (2s)

^M (s) .

We may ^{remark in} passing ^that this formula provides ^a fermionization of the Riemann gas. ^{Let us}

rewrite the above equation ^as

1374

the second factor on the right ^{hand side can be} reinterpreted ^{as the} partition function of a Riemann gas ^{at the same} temperature but with double energies. ^We may iterate and find

The higher energy levels should be irrelevant at low temperature (this ^{of course} requires ^a

careful study). ^{So we} may speak ^{of an} equivalence between fermions (an ^infinite family ^{of Môbius} gases) and bosons (the ^Riemann gas).

More generally ^the partition ^{function of a} statistical system is the "extended" Laplace ^trans-

form of the density ^{of states :}

"extended" because the range ^of integration may ^include negative values of the energy. ^{Let us}

note that this is not quite the Mellin transform which has a strict definition in number theory :

it relates the (additive) generating ^functions typically ^some modular forms to the (multiplicative) generating ^functions typically ^{zeta functions or} so-called L-functions. The general formula is :

G (s) is Euler’s Gamma function which is the "bridge" ^between multiplicative ^{and additive} aspects

of number theory (it ^{is a} product of consecutive numbers ! ). ^In components

and D (s) _ ^{is as above.}

We have not yet given ^the precise definition of a "multiplicative" arithmetic function : it is

a function such that its value at the product ^{of two} relatively prime integers ^{is the} product ^of

its values at these integers. ^This property is stable under Môbius transform. More generally ^it

is stable under the "tensor product" ^of systems i.e. under product ^{of the} generating (partition)

functions. So called Hecke modular forms have multiplicative ^Mellin transforms, ^{and we have} given ^a few examples ^of generating series with multiplicative coefficients, zeta, its inverse etc... A

"strong" ^{form of} "multiplicativity" should be translated as absence of interactions between primes.

It corresponds ^{to the existence of an Euler} product decomposition ^{over the} primes. ^{But we should}

not restrict ourselves to free systems ! t And Dirichlet series of both types ^exist independently ^of

the additivity ^or multiplicativity of their coefficients and we have seen that they ^are quite generally Mellin-Laplace trasnforms of each other. The physical interpretation ^{of these} generating ^functions

however depends ^{on these} properties.

1b conclude this announcement we would like to add four remarks. The advantage ^{of these}

models of arithmetical statistical mechanics is that the energy ^{levels are} exactly ^{known at least}

in principle, ^{we are} left with the purely physical problem ^of studying thermodynamical quantities typically Log ^D (s) ^{which is} singular ^at phase transitions Lee-Yang [4].

The Euler function admits also a kind of parton decomposition ^{à la} Weierstrass, ^more gener-

ally any Euler (over ^the primes) ^{or more} general product formula leads to such a decomposition

of Dirichlet series.

The Riemann hypothesis ^{admits a nice} formulation in terms of Môbius fermionic ghosts. ^It

is equivalent ^to saying ^{that the} integrated density ^{of states of that gas} is less than a constant times

so-called xi function (or generalization thereof) ^with singularity

could heat the system ^{all the} way ^{to the double} temperature s

1/2.

Acknowledgements.

I would like to thank the organizers of the winter school at les Houches who allowed me to

present ^this general scheme. 1 learnt there that Dr J.H. Hannay ^{and Dr D.} Spector ^{had made}

similar remarks in some special ^cases independently ^{but that} apparently they ^{did not} consider the

general picture ^nor phase transitions. 1 also leave for a future paper ^the comparison ^{with the}

rather different semi-classical approach ^of quantum ^chaotic billiards, these have periods multiples

of Log p’s and relate the zeroes of zeta to the values of their "physical" energies.

References

[1] ^{SERRE J.P., cours} d’Arithmétique (PUF, Paris) ^1970.

[2] ^RIEMANN B., Monatsberichte der Berliner Ak. November (1859).

[3] ^{HAGEDORN R.,} Suppl. Nuovo Cimento 3 (1965) ^{147 ;}

see also RUMER Yu.B., J. Exp. ^Theor. Phys. ³⁸ (1960) ^1899.

[4] LEE T.D. and YANG C.N., Phys. ^{Rev. 87} (1952) ^404.