ON GENERAL ZETA FUNCTIONS

Transcription

1 ON GENERAL ZETA FUNCTIONS P. DING, L. M. IONESCU, G. SEELINGER Abstract. An abstract interface to zeta functions is defined, following the Lefschetz-Hasse-Weil zeta function as a model. It is implemented in terms of path integrals with the statistics physics interpretation in mind. The relation with Riemann zeta function is explained, shedding some light on the primon gas. Contents 1. Introduction 1 2. An interface to zeta functions An arithmetic interface to zeta functions Euler product form The trace / determinant interpretation A Path Integral Interpretation 5 3. Relation with Hasse-Weil zeta function 6 4. The relation with Lefschetz Zeta Functions 7 5. The relation with Dynamical Zeta Functions 7 6. The Relation with Riemann Zeta Function Relation with Algebraic Quantum Groups 8 7. Primon Gas as a Fynmann Path Integral Quantum System 8 8. Conclusions and further developments 9 References 9 1. Introduction To understand prime numbers, as one of the most pressing task in mathematics, the structure of basic finite fields can be probed using the Reidemeister torsion for groups [1]. To better understand generating functions of the zeta function type, an interface for Hasse-Weil zeta function [4] is extracted in 2. Date: September 6,

2 2 P. DING, L. M. IONESCU, G. SEELINGER It is then compared with Lefshetz-Artin-Mazur-Ruelle zeta functions, and finally with the Riemann zeta function, in the context of statistical mechanics of the primon gas. 2. An interface to zeta functions We first extract from [4] the general properties of the Hasse-Weil zeta function. Having in mind applications to primon gas [8] (statistics of number theory [9]), the formalism of the zeta function is tailored to fit within the (Feynman) path integral formalism, so that the zeta function becomes the partition function of the associated quantum system. We first separate the arithmetic interface to zeta functions of Hasse- Weil type, taking as model the Hasse-Weil zeta function in algebraic geometry An arithmetic interface to zeta functions The zeta function of a POSet. Let deg : X (N, ) be a degree function, where (N, ) is the lattice of natural numbers, with divisibility relation as a partial order. Remark 2.1. Here X corresponds to the union of solutions of a polynomial equation over finite field extensions defining an affine variety, as described in 3. The set X is partitioned into disjoint sets X n = deg 1 (n), n N, with a n = X n elements each. The pull-back on X of the lattice structure of (N, ) is denoted <<. The pull-back of the cardinal measure on N through deg is denoted dm: dm = deg(x)dx, where dx is the counding measure on X. Then deg : (X, <<, dm) (Z,, card) is a morphism of measure spaces with lattice structures with minimal element 1 N. The number of points with multiplicity is the arithmetic function N(n) = d n Deg(X d ) = d X d = d n d a(d), (1) where Deg(X d ) = d X d is the total degree (mass) of X d.

3 ON GENERAL ZETA FUNCTIONS 3 Remark 2.2. If we define U n = d n X d then we obtain a directed family of sets, and the number of points with multiplicity is represented as an integral of the degree function on the directed family: N(n) =< U, deg > (n) = dm = deg(x)dx. d n X d U n In the algebraic geometry case the starting point for defining the Hasse- Weil zeta function 3. Linearize the degree function on the free module of divisors generated by the set X: deg : ZX Z, deg( n i x i = n i deg(x i ). Definition 2.1. The zeta function (ZF) of the POSet (X, deg, <<) is Z(X, t) = exp( n N N(n)t n /n). (2) Remark 2.3. We would like to emphesize the roles of various factors in the above definition. The power t n defines the grading. The factor 1/n together with the cardinal measure dn corresponds to the usual integration measure d ln t of Mellin transform. Remark 2.4. The overall phylosophy of zeta functions as generating functions is that the this zeta function of the object X has the form: Z X (t) = exp( G X (t)dt), G X (t) = a n t n n 0 with G X (t) the generating function of the sequence of numbers {a n } n N, i.e. the combinatorial way of switching between the completion of the group algebra of formal power series to the convolution algebra of arithmetic functions for a given monoid Rationality of zeta function. In general, when X is finite the zeta function is a rational function (following [4], p.5). Proposition 2.1. Z(X, t) = x X 1 1 tdeg(x). (3) Proof. lnz(x, t) = N(n)t n /n = d a(d)t n /n = t md /m n N n N d N m 1 d n

4 4 P. DING, L. M. IONESCU, G. SEELINGER where equation 1 for N(n) was used, and followed now by a formal manipulation of power series, with the usual logarithmic power series, the inverse of the universal generating series we denote by e x 1 : = ( a(d))ln(1 t d ) = [ ] ln(1 t d ) a(d) = ln (1 t d )) a(d). d 1 d 1 d 1 There is no issue of convergence in the sense of functions: we are concerned with the algebraic structures of formal series; but if we insist, the convergence is n-adic, as usual in the context of graded rings. The rational form of the zeta function Z(X, t) (RHS / Euler s product form) follows by collecting expanding the product over X = d 1 X d : t d 1(1 d ) a(d) = (1 t deg(x) ). x X Remark 2.5. The proof should be reinterpreted in the context of convolution arithmetic functions, probably hiding the Mobius inversion formula. Remark 2.6. The relation with DFT of deg : Z/NZ N, where N = X, should also be explored (Generating Functions [], Discrete and Continuum Book []??); the RHS product form is a DFT of convolutions of step functions (characteristic functions of X d ), while the divisors d N suggest the use of Chinese Remainder Theorem (local to global principle) Euler product form. The above product form is called the Euler product form of the zeta function. In other cases it is the starting point of defining the zeta function, e.g. Ihara zeta function etc The trace / determinant interpretation. In many cases such a zeta function comes from a counting trace [?], p.9. Let (M, f) be a dynamical system. We only state the main formulas, without concern for convergence, regarding the equations from an algebraic point of view of formal series (convolution algebras). If Tr : (F(M), +) k is a trace on some algebra of operators on M, i.e. linear such that Tr(L L ) = Tr(L L) then the associated zeta 1 e x = n N xn /n! generalizes to groupoids [10].

5 ON GENERAL ZETA FUNCTIONS 5 function is a determinant (multiplicative): N(m) = Tr(L m ), ζ L (z) = exp( N m z m /m) = det(i zl), where I is the identity operator. Then the rational form (Euler product) comes from a spectral representation of the determinant in terms of eigenvalues (factorization of the characteristic polynomial). For example if L = f acting on homology of M, then Lefschetz trace formula leads to Lefshetz zeta function A Path Integral Interpretation. In view of the path integral application, we start from the statistical mechanics interpretation of a zeta function as a partition function. A local zeta function Z(t) of the Lefschts-Hasse-Weil type is the logarithmic derivative of the generating function F(t) of the arithmetic function N : N C [6] Z(t) = exp( N(n)t n /n), F(t) = d/dt lnz(t) = N(n)t n. When interpreting the zeta function as a partition function, F(t) is the thermodynamic energy [7]. Relation with energy levels The logarithm of the Euler product reveals the energy levels???: lnz(t) = ln(1 t deg(x). OR via RZF: Mellin substitution t < > e s AND deg = ln, i.e. the degree function associates to an element x its divisor X x in g X We use the lattice structure to define a path space structure in N: γ n N γ : 1 n, where the notation γ replacing d, suggests the path interpretation of the divisor d of n. Moreover is interpreted as the action 2. 1 S(γ) = lna(d) 2 The general calculus suited for quantum theory is the path integral formalism.

6 6 P. DING, L. M. IONESCU, G. SEELINGER Then the number of points function is the propagator: N(n) = e S(γ) = K(1, n).??? path space in X: γ:1 n γ : x y iff deg(x) deg(y) 3. Relation with Hasse-Weil zeta function In the case of an affine variety X = {x F p f(x 1,, x l ) = 0}, for some polynomial f F p [X], we distinguish two levels of generality, when considering closed points X cl or more general K-points. The closed points case originates with finite fields in characteristic p. The number of elements : FF p (N, ) is a complete invariant. We will fix a section F p (n) = F p n associating to n N the unique modulo isomorphism finite field with p n elements: F p : (N, ) FF p, F p (n) = n. Below we show the diagram relating finite fields in characteristic p and finite sets, where k = F p. X cl FF p Sets X F p (N, ) (N, +, card) ν p a Here X cl (F q ) is the set of solutions in F q of f(x) = 0 with f k[x]. The degree function deg(x) = [F p (x) : F p ] is defined as the degree of the field extension obtained by adjoining the element x F q. If q = p n for some n N, then the extension [F q : F p ] has degree n and we must have deg(x) n. Thus the degree function is compatible with the directed system X cl, as well as with the directed set X. With the notation a(n) = X n (recall X n = X(F p (n))) Deg(X n ) = x X n deg(x) is the total degree at level n. The setup of K-points represents the category of finite fields via Hom(K, ), where K is a field of the same characteristic p. It leads to

7 ON GENERAL ZETA FUNCTIONS 7 a functorial mapping X : (POSet, <<) Ab into an abelian category, for example [4] in the case of an affine algebraic variety: X FF R p Ab F p X (N, ) where the K-points functor is X R = Hom k alg (R, ) for some commutative ring R. 4. The relation with Lefschetz Zeta Functions 5. The relation with Dynamical Zeta Functions 6. The Relation with Riemann Zeta Function As stated in [4], Remark 3.3.,, p.5, the relation with RZF is via interpreting the LHW-zeta function as a generating function of the effective 0-cycles (divisor), i.e. elements of g X = ZX. Given such a 0-cycle / divisor α = n i x i, with n i 0 Z(X, t) = x X(1 + t deg(x) + t 2deg(x) + = α g X t deg(α). If we apply this to the case X = P and deg(p) = lnp, then g P = ZP, and we obtain the Riemann zeta function, as stated above (see Diagram 4): Z(P, t) = P α(p)ln p = t lnn = n s = ζ(s) n N α ZP + t where α = p P α(p)p (formal sum), n = Exp(α) = p α(p), together with the formal Mellin substitution t lnn = n s 3. 3 Here one silently passes from formal series, in the context of convolution algebras, to real parameters when convergent; then, extended via analytic continuation, when daring towards Pontryagin, Fourier and Mellin duality. n N

8 8 P. DING, L. M. IONESCU, G. SEELINGER 6.1. Relation with Algebraic Quantum Groups. What we need at this stage is a correspondence between the complex analysis rhealm of Riemann-Cauchy exponentials p s and the multiplicative characters of an algebraic quantum group, within the algebraic rhealm of group / convolution algebras, which should correspond to the additive characters t k. This should yield a functorial correspondence between Foruier additive duality and Mellin multiplicative duality formulated as duality of algebraic quantum groups [5]. 7. Primon Gas as a Fynmann Path Integral Quantum System By setting X = P = Spec(Z), x = p, and deg(x) = lnp, t = e s, the above product is analogous to the Euler product to the Riemann zeta function: Z(X, exp( s)) = 1 1 p = n s = ζ(s). s p P So, our goal is to relate the above formalism of Leipchitz-Hasse-Weil zeta function, with the statistical mechanics formalism, via the primon gas (Riemann gas) [?]: Energylevels : E p = lnp, Occupation numbers : n = exp( e(p)lnp (divisor : D = e(p)x p ), Partition function : Z(s) = exp(se n ), (s = 1/kT). The divisibility lattice should yield a Feynman path integral formalism, which in turn will yield the partition function as a Feynman Path Integral (FPI): FPI : Z(s) = K(A, B) F Amplitude : K(A, B) = A,B Obj(C) γ Hom(A,B) exp( S(γ)), S(γ) = γ L(p, q), where C is the category of states A Obj(C) and transitions γ (paths) between states Hom(A, B). Remark 7.1. The Hilbert-Polya historic suggestion of realizing Rymann Hypothesis as a self-adjoint condition in a quantum mechanics model should be upgraded in view of the success of FPI quantization, String Theory and Riemann surfaces relation, and Kontsevich s tools for proving Formality Theorem (byproduct: exitence of star products, a.k.a of

9 ON GENERAL ZETA FUNCTIONS 9 deformation quantizations of Poisson manifolds / symplectic mechanics). When the POSET of primes is endowed with the natural measure dm = lnpdp, as in the functional model of rationals r = p P pk(p) : g = ZP exp (Q, ) R k(p)dp ln R the above proof still holds for the Riemann zeta function, with its interpretation as a LHW-zeta function. ** PROOF? ** (4) 8. Conclusions and further developments References [1] L.M. Ionescu, Reidemeister torsion and finite fields, work in progress. [2] Alexander Fel shtyn, Dynamical zeta functions, Nielsen theory and Reidemeister torsion, chao-dyn/ v2, [3] L. Ionescu, The POSET of prime numbers, in preparation. [4] M. Mustata, Lectures on Hasse-Weil zeta function, mmustata/lecture2.pdf [5] A. van Daele, The Foruier transform in quantum group theory, scaenepe/vandaele.pdf [6] zeta function [7] Wikipedia, Partition function (statistical mechanics), function (statistical mechanics). [8] Wikipedia, Primon gas. [9] B. Julia, Statistics of natural numbers (?) [10] Categorical generating functions: yoshida, Baez: Grioupoidification made easy, M. Haiman Department of Mathematics, Illinois State University, IL address: lmiones@ilstu.edu