A lattice gas of prime numbers and the Riemann Hypothesis
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] J un A lattice gas of prime numbers and the Riemann Hypothesis
Fernando Vericat
Grupo de Aplicaciones Matem´aticas y Estad´ısticas de la Facultad deIngenier´ıa (GAMEFI). Universidad Nacional de La Plata, Argentina ∗ Abstract
In recent years, there has been some interest in applying ideas and methods taken from Physicsin order to approach several challenging mathematical problems, particularly the Riemann Hy-pothesis. Most of these kind of contributions are suggested by some quantum statistical physicsproblems or by questions originated in chaos theory. In this article we show that the real part ofthe non-trivial zeros of the Riemann zeta function extremizes the grand potential correspondingto a simple model of one-dimensional classical lattice gas, the critical point being located at 1/2as the Riemann Hypothesis claims.
PACS numbers: 02.10.De, 05.20.Gg, 05.50.+qKeywords: Lattice gas, prime numbers, variational principle, Riemann Hypothesis ∗ Also at Instituto de F´ısica de L´ıquidos y Sistemas Biol´ogicos (IFLYSIB)-CONICET-CCT La Plata, Ar-gentina.; E-mail: vericat@iflysib.unlp.edu.ar . INTRODUCTION The relation between physics and number theory has a long and fruitful history as thetwo books referenced [1] and [2] suggest. This link is in both directions: from numbertheory to physics and from physics to number theory. At the beginning, the first directionwas the more frequently transited and so many number theoretic functions (and conceptsin general) were used in field theory, in classical and quantum statistical physics and alsoin chaos theory. In this sense the Riemann zeta function[3] defined by ζ ( s ) := P ∞ n =1 n − s inthe half-plane ℜ ( s ) > ζ ( s ) in physics as well as in mathematics itself, its behavior,particularly the properties of its zeros, has been much studied. Nowadays we know that ζ ( s ) = 0 has as trivial solutions the negative integers − k with k ∈ N . The so-called non-trivial zeros are complex numbers most of whose properties are also known. In an applicationof ζ ( s ) to a number theoretic problem, specifically the distribution of the prime numbers,Riemann conjectured more than 150 years ago that all the complex zeros have 1/2 as realpart[3]. This statement is the famous Riemann Hypothesis whose proof has unsuccessfullyinvolved many mathematicians since then. This apparent inaccessibility to their proof from afull rigorous mathematical point of view has, perhaps, motived the interest in applying ideasand methods taken from physics in order to approach the Riemann Hypothesis in particularas well as several other number theoretic problems. In this way we can mention the workdone around the idea of Hilbert and Polya, proposed at the beginning of the past century,that the complex zeros of ζ ( s ) constitute the spectrum of an operator R = 1 / I + i H where the Hamiltonian H is self-adjoint. Also, it is worth mentioning diverse applicationsof statistical physics such as the use of random walks in order to approach the RiemannHypothesis[4],[5] or the search for statistical mechanics models whose partition functionsbe related to ζ ( s ) by using spin chains[6] or gases of harmonic oscillators[7]. These (andseveral others) contributions of physics to number theory can be seen in the excellent reviewby Schumayer and Hutchinson[8].In this article, following the way from physics to number theory, we give support to theRiemann Hypothesis by extremizing the grand potential of a simple classical one-dimensionallattice gas of prime numbers. In next section we discuss the model and the equilibrium2tatistical mechanics functions that we need for our analysis, particularly the grand potentialof the system. Section 3 is devoted to present the explicit formula given by Riemann forthe prime-counting function in terms of the zeros of the zeta function and, as a naturalconsequence of the enunciation of the properties of these ones, to state again the RiemannHypothesis. Finally in section 4 we give support to the Riemann Hypothesis from our modelby showing that the real part of the complex zeros of the zeta function must extremize thegrand potential obtained in section 2. Three appendices are considered in order to justifysome equations of the main text.
2. LATTICE GAS OF PRIME NUMBERS
Physically our model is very simple: a one-dimensional lattice gas in the grand canonicalensemble. The lattice is an interval of the natural numbers: [1 , M ] ⊂ N where M is largeenough (eventually, in the thermodynamic limit, M → ∞ ). The system is in contactwith a particles reservoir characterized by the chemical potential µ and a heat reservoir attemperature T . For N particles the configuration of the system is given by ω ≡ ( ω N , N )where ω N ≡ ( i , i , · · · , i N ). The coordinates i α ( α = 1 , , · · · , N ) take values in [1 , M ] and N ranges, in principle, between 0 and ∞ . The set of such configurations, the configurationspace, will be denoted Λ.We assume that each site in the lattice can have at most one particle, so we are consideringa hard point pair potential u ( i α , i β ) = (cid:26) ∞ if i α = i β i α = i β . (1)Also we assume that the particles are subject to a one-point potential such that they canjust occupy sites in the lattice which are prime numbers: u ( i α ) = (cid:26) i α is prime ∞ if i α is composite. (2)By the way, an example of explicit function with this form is[9]: u ( i α ) = − log (cid:2) sin (cid:8) π [( i α − /i α (cid:9) / sin { π/i α } (cid:3) . The energy for the configuration ω is u ( ω ) = P Nα =1 u ( i α ) + P Nα<β u ( i α , i β ).3athematically, a state of the system is a probability vector ν = ( ν ( ω ) | ω ∈ Λ ). Theset of all states is denoted by M . In the state ν the system has the mean energy ν ( u ) := P ω ∈ Λ ν ( ω ) u ( ω ). The (grand) partition function of u is defined Ξ ( β, µ ) := P ω ∈ Λ exp [ − β ( u ( ω ) − µN )] where β = ( k B T ) − with k B the Boltzmann constant. Forthe parameters β, M and µ the Gibbs measure is defined ν ( ω ) := 1Ξ ( β, M, µ ) exp [ − β ( u ( ω ) − µN )] . (3)Given a measure ν we have the entropy H ( ν ) := − P ω ∈ Λ ν ( ω ) log ν ( ω ) and the grandpotential Ω [ ν ] := ν ( u ) − β H ( ν ) − µν ( N ) (4)with ν ( N ) := P ω ∈ Λ ν ( ω ) N .To introduce the notion of equilibrium state we consider a variational principle for finitesystems ( | Λ | =finite) according to which, for given energy u and parameters β, M and µ , theGibbs measure satisfiesΩ [ ν ] = − β log Ξ ( β, M, µ ) = inf ν ∈M (cid:18) ν ( u ) − β H ( ν ) − µν ( N ) (cid:19) . (5)A measure that attains this infimum is called an equilibrium state. Gibbs measure ν is thusan equilibrium state. The result given by Eq.(5) is easily demonstrated[10] by using Jenseninequality applied to the concave function x ln x (see Appendix A). The principle can beexpressed saying that for any measure ν is Ω [ ν ] > Ω [ ν ] = − β log Ξ ( β, M, µ ) with equalityif and only if ν = ν .Turning to our model, the grand potential for the Gibbs state is easily calculated if weexplicitly write the grand partition function ( z = exp [ βµ ]):Ξ ( β, M, µ ) = ∞ X N =0 z N N ! M X i =1 M X i =1 · · · M X i N =1 exp " − β N X α =1 u ( i α ) + N X α<β u ( i α , i β ) ! and observe that, because of the limitation of occupation to just the prime numbers (Eq.2),the coordinates i α ( α = 1 , , · · · , N ) can take values only among the π ( M ) prime numbersthat exist in the interval [1 , M ]. Also, because of the impenetrability of the particles (Eq.1),the choice can be made of π ( M )! / [ π ( M ) − N ]! manners. We obtain4 ( β, M, µ ) = π ( M ) X N =0 (cid:18) π ( M ) N (cid:19) z N = (1 + z ) π ( M ) (6)and Ω [ ν ] = − β log (1 + z ) π ( M ) . (7)
3. RIEMANN HYPOTHESIS
The prime-counting function π ( x ) is the number of prime numbers less or equal than x .In his 1859 classic memoir to the Berlin Academy of Sciences (see for example the monographby Edwards[3] for a translation), Riemann gave an explicit formula for π ( x ) in terms of thezeros and the pole of the analytical continuation of ζ ( s )[3]: π ( x ) = π ( x ) + e π ( x ) , (8)where the smooth part, to which contribute the pole at s = 1 and the integer zeros of ζ ( s ),is given by π ( x ) = ∞ X n =1 µ ( n ) n " Li (cid:0) x /n (cid:1) − ∞ X k =1 Li (cid:0) x − k/n (cid:1) , (9)with Li( x ) the logarithmic integral function and µ ( n ) the M¨obius function defined µ ( n ) = n has one or more repeated prime factors( − r if n is a product of r distinct primes1 if n = 1 . The complex zeros, on the other hand, contribute to the oscillatory part e π ( x ) = − ∞ X n =1 µ ( n ) n ∞ X α =1 Li (cid:0) x ( σ α + it α ) /n (cid:1) , (10)where the real numbers σ α and t α are the real and imaginary part, respectively, of thecomplex number ρ α = σ α + it α that verifies ζ ( ρ α ) = 0 . x large enough the non-oscillatory and oscillatory parts can be written, respectively,as π ( x ) ≈ x log x (11)and (see Appendix B) e π ( x ) ≈ − x ∞ X α =1 x σ α ( σ α + t α ) [ σ α cos ( t α log x ) + t α sin ( t α log x )] . (12)For simplicity, from now on we will consider this last situation; so M will be assumed largeenough.As we mentioned in the Introduction, the properties of the complex or non trivial zerosof the Riemann´s zeta function has been extensively studied[3]. For example it has beendemonstrated that there are infinitely many complex zeros and that all of them lie inside theregion 0 < ℜ ( s ) < ℑ ( s ) = 0. Also is well knownthat if ξ ( s ) := s ( s − π − s/ Γ (cid:0) s (cid:1) ζ ( s ) then the non trivial zeros of ζ ( s ) are preciselythe zeros of ξ ( s ) and since ξ ( s ) = ξ (1 − s ), we have that the complex zeros of ζ ( s ) aresymmetric with respect to the so called critical line ℜ ( s ) = 1 /
2. The Riemann Hypothesisis the statement that all the complex zeros have their real part exactly on the critical line: σ α = 1 / ∀ α . Although billions of complex zeros have been numerically calculated[11]confirming all of them this conjecture, the full demonstration of the general validity of theRiemann Hypothesis remains still open.
4. SUPPORTING THE RIEMANN HYPOTHESIS FROM THE MODEL
We wish to analyze the behavior of the equilibrium grand partition potential (Eq.7), with π ( M ) given by Eqs.(8-12), as a function of the quantities σ α ∈ (0 ,
1) ( α = 1 , , · · · ). Thesenumbers must be thought as possible values for the real part of the zeta function zeros andour goal would be to identify the true ones.We focus into a generic zero ρ γ and his complex conjugate ρ ∗ γ (which is known is alsoa zero) and observe that, since Riemann actually didn’t know the reliable values of thecomplex zeros, Eqs. (10) and (12) must be taken as general enough as to contemplate thepossibility that, associated to t γ = ℑ ( ρ γ ), both cases, σ γ = 1 − σ γ and σ γ = 1 − σ γ , can beconsidered for ℜ ( ρ γ ). 6xplicitly we rewrite Eq.(7) for β, M and µ fixedΩ ( β, M, µ ; σ γ ) = − β log (1 + z ) π ( M ; σ γ ) , (13)with π ( M ; σ γ ) = π ( M ) + e π = γ ( M ) + e π γ ( M ; σ γ ) . (14)Here the term e π = γ ( M ) includes all the complex zeros except those labelled γ and we assumethat its contribution to the prime-counting function is the true one. We write the remainingterm: e π γ ( M ; σ γ ) ≈ − M (cid:16) M σγ σ γ + t γ [ σ γ cos ( t γ log M ) + t γ sin ( t γ log M )] + M ( − σγ )(1 − σ γ ) + t γ [(1 − σ γ ) cos ( t γ log M ) + t γ sin ( t γ log M )] (cid:17) for σ γ = 1 − σ γ − M M / + t γ (cid:2) cos ( t γ log M ) + t γ sin ( t γ log M ) (cid:3) for σ γ = 1 − σ γ , (15)where we have taken into account the fact that if ζ ( σ γ + it γ ) = 0 then also ζ (1 − σ γ + it γ ) =0, ζ ( σ γ − it γ ) = 0 and ζ (1 − σ γ − it γ ) = 0.Special cases can make the analysis even simpler. For M = [ M ] ([ • ] integer part of • )with M = exp[ π (4 n − / | t γ | ] and M = [ M ] with M = exp[ π (4 n − / | t γ | ] ( n ∈ N largeenough) and taking into account that | t α | > . e π γ ( M ; σ γ ) ≈ ± | t γ | log M , (cid:16) M σ γ , + M (1 − σ γ )1 , (cid:17) for σ γ = 1 − σ γ ± | t γ | log M , M / , for σ γ = 1 − σ γ , (16)where the upper sign corresponds to M . Note the avoidable discontinuity in the function e π γ ( M ; σ γ ) when σ γ = 1 − σ γ .Using Eq.(13) together with Eq.(15) -or the particular cases given by Eq.(16)- is easy tosee that the grand potential has a unique extremum (minimum or maximum depending on M ) at the interval 0 < σ γ <
1. From a heuristic point of view it is reasonable to expectthat this extremum (maximum = unstable equilibrium; minimum = stable equilibrium)7e reached when σ γ takes the value ( σ γ ) true that gives, through Eq.(14), the correct time-invariant number of prime numbers lying inside the interval [1 , M ].The same conclusion can be achieved more formally by considering the problem from adynamic point of view. To this, we firstly take into account that fixing a given value for σ γ can be thought as imposing a constraint that limits (through π ( M ; σ γ )) the number of theaccessible states to the system. If this constraint is removed, so σ γ is leaved to freely varyin the interval (0 , P ( σ γ ) that the system be in stateswith the parameter taking values in the interval between σ γ and σ γ + δσ γ behaves as (seeAppendix C) P ( σ γ ) ∝ (cid:26) exp [ − β Ω ( β, M, µ ; σ γ )] if σ γ ∈ (0 , σ γ / ∈ (0 , . (17)We can then arbitrarily introduce a second random variable such that (17) be the marginalof the joint probability of both ones. We use as new variable a Maxwellian one · σ γ so thatthe bi-dimensional random variable (cid:16) σ γ , · σ γ (cid:17) describes the phase space of an hypotheticalparticle of mass m moving in a force field with potential function given by U ( σ γ ) = (cid:26) Ω ( β, M, µ ; σ γ ) if σ γ ∈ (0 , ∞ if σ γ / ∈ (0 , . (18)In this picture, P ( σ γ ) can be thought as the marginal of the joint probability distribution P ( σ γ , · σ γ ) = 1 Z ( β ) exp (cid:26) − β (cid:20) U ( σ γ ) + 12 m · σ γ (cid:21)(cid:27) . (19)Here the normalization constant Z ( β ) is the single particle canonical partition function Z ( β ) = Z Z e − β H (cid:16) σ γ , · σ γ (cid:17) dσ γ d · σ γ , (20)where H (cid:16) σ γ , · σ γ (cid:17) denotes the Hamiltonian H (cid:16) σ γ , · σ γ (cid:17) = 12 m · σ γ + U ( σ γ ) . (21)The Hamilton equations of motion for the variables σ γ and · σ γ are · σ γ = ∂ H m∂ · σ γ , m ·· σ γ = − ∂ H ∂σ γ , (22)which yield the Newton equation 8 · dσ γ dt = − dU ( σ γ ) dσ γ . (23)If we assume that initially the particle is placed at the position σ γ = ( σ γ ) extr thatextremizes Ω ( β, M, µ ; σ γ ) with velocity · σ γ = 0, then the equation of motion has as theunique obvious solution the isolated fixed point σ γ ( t ) = ( σ γ ) extr ∀ t . Taking into accountthat for any other pair of initial conditions the solution is time-dependent and that in ourlattice gas model the number π ( M ; σ γ ) of prime numbers less or equal than M does notchange with time but remains constant, we deduce that it must be ( σ γ ) true ≡ ( σ γ ) extr . Onethen infers that ( σ γ ) true should verify ∂ Ω ( β, M, µ ; σ γ ) ∂σ γ (cid:12)(cid:12)(cid:12)(cid:12) σ γ =( σ γ ) true = − β log (1 + z ) ∂ e π γ ( M ; σ γ ) ∂σ γ (cid:12)(cid:12)(cid:12)(cid:12) σ γ =( σ γ ) true = 0 . (24)Derivation of expressions (15) or (16) shows that this equation has the form[ f ( σ γ ) − f (1 − σ γ )] σ γ =( σ γ ) true = 0 (with f ( σ γ ) = 0; f (1 − σ γ ) = 0) which is clearly ver-ified by ( σ γ ) true = 1 − ( σ γ ) true = 1 /
2. Note that the maximum (minimum) achieved at 1/2when the discontinuity is avoided is smaller (larger) than the corresponding values taken bythe function e π γ ( M ; σ γ ) in (15) or (16) when σ γ = 1 − σ γ .Because the zero ρ γ that we have considered is a generic one, we would conclude that( σ α ) true = 1 / ∀ α , say that, in fact, the real part of all non-trivial zeros of Riemann’s zetafunction lie on the critical line. Acknowledgements