AN EXPLORATION OF THE PRIME NUMBERS

Transcription

1 AN EXPLORATION OF THE PRIME NUMBERS LUCIAN M. IONESCU Abstract. A natural lattice structure on the set of prime numbers is derived from the internal symmetries of the corresponding primary finite fields. It leads to a correspondence with the Hopf algebra of rooted trees, and as an application, to an alternative approach to the Prime Number Theorem. Additional tools are introduced, exploring the quantum groups approach to multiplicative number theory, from a categorical perspective. Contents 1. Introduction 1 2. A Partial Order on The Set of Prime Numbers The Lie-Klein picture of a finite fields The complexity of primes Gradings on P 3 3. Prime Numbers and Rooted Trees 4 4. The rational numbers as a formal group An external composition on prime numbers The rationals as a formal group Further avenues of exploration 6 5. The Prime Number Theorem An analog of Riemann-Roch Theorem for Q Breaking Down The PNT A probabilistic framework On Chebyshev s estimates Conclusions 10 References Introduction There is a need for a new insight into the ring of integers [2], p.1, [3], [4], p.143, etc. As usual, physics comes to the rescue 1 : in a quantum digital universe, the primary modes of vibration (periodic structures) should be modeled by primary fields F p, reflecting the value of the fine structure constant via a modern, quantum computing, model of the hydrogen atom [1]. Date: September 1, E.g. invariants of knots, 4-manifolds etc. 1

2 2 LUCIAN M. IONESCU Moreover, there is a wide belief that Number Theory is the ultimate physics theory [5]; [6]: It is remarkable that the deepest ideas in number theory reveal a far-reaching resemblance to the ideas of modern theoretical physics. One way or another, categorification is the correct approach: instead of investigating sizes of objects, study the objects themselves, not their shadows : Categories : Finite Sets S Simple objects Spec S Grothendieck Tangent mapping (generators) Number Systems : Integers Z Prime Numbers Spec(Z)). Without loss of generality for modeling reality, one can restrict to the study of the category of finite sets S. Once a coordinate system is chosen, reducing their symmetry groups, the category of finite abelian groups is obtained A. The main object of study, then, is their symmetries, the internal functor Aut A. In this article we will introduce a lattice structure on the set of primes, derived from Aut A, leading to a mapping into the Hopf algebra of rooted trees. It is natural to investigate what are the functions analog to those involved in the proof of the Prime Number Theorem, and whether on can obtain a better understanding of the latter lifting it through the canonical POSet morphism from the lattice of primes, to natural numbers. In conclusion, the connection between these two primeval structures, primes and trees, suggests deep results in the light of Selberg s trace formula interpretation of the Riemann-Mangoldt/Weil exact formula. Additional explanations regarding the author s approach justifying some of the above claims will be given in a follow up article. 2. A Partial Order on The Set of Prime Numbers 2.1. The Lie-Klein picture of a finite fields. The automorphisms of (F p, +) as an abelian group (Z-module) form a group isomorphic to the corresponding multiplicative group via the regular representation. Its summands are elementary factors of the form Z q k, corresponding to the factorization of Euler s totient function value: L : (F p, ) Aut(F p, +) = p PZp k(p), φ(p) = p 1 = p k(p). Here a positive integer is view as a function on the set of primes P. 2 For example, Aut(F 19 ) = Z 2 Z 3 2, since 19 1 = The partial factorization of φ(p) = p 1 = 2 k q will be called the Proth factorization The complexity of primes. In this way the smaller primes q dividing p 1 represents the Lie generators of the space (F p, +), allowing for the definition of a natural partial order. Definition 2.1. A prime q is called a generator of the prime p, denoted q << p iff q divids p 1. p P 2... towards a function field interpretation of the rationals, and application of duality of algebraic quantum groups.

3 AN EXPLORATION OF THE PRIME NUMBERS 3 For example 2 and 3 are generators of p = 19. Fermat primes p = 2 n + 1 have only one generator, q = 2, if disregarding its multiplicity n. Proposition 2.1. The set of prime numbers (P, <<) is a connected partial ordered set (POSet) with minimal element 2. The identity function: I : (P, <<) (N, <), I(p) = p is a morphism of POSets, representing a refinement of the total ordering of primes by size. Remark 2.1. This POSet is probably not a lattice. The Euclid s trick invites for defining p p, but there seems to be no reason for the uniqueness of p p, their common symmetries Gradings on P. There are a few natural functions defined on P, including a grading by the number of their symmetries. Definition 2.2. The function w : P N, w(p) = dim(aut(f p, +)) is called the grading of primes by complexity. The term support is used since the author interprets the multiplicative rationals (positive), as functions on P: r = m/n : P Z, corresponding to their unique factorization. For example w(19) = 2, since the symmetries of F 19 has two generators (the rank, or local dimension, of the Z-module is two). Alternatively, p 1 = 19 1 = 18 is a function supported on two points: 2 and 3. The reader interested in the long-term research program of the author is invited to consider the following justification. Remark 2.2. Classical algebraic geometry evolved with Grothendieck in the abstract, categorical theory of schemes. Modern mathematics, especially the russion school, supported by meaningful applications to physics, developed rather in the direction of deformation theory, as a natural continuation of the ideologies of Felix Klein and Sophus Lie [13]. Also, analysis, as a contructive approach to mathematics, faded away in favor of the algebraic, top-down approach in designing mathematical theories. Therefore, the author inclines towards a quantum groups approach to p-adic numbers, which will be shown to be just a graded approach to (discrete) mathematics, rather then mathematics in characteristic p [8]. Another reason for branching in a new direction, is that the duality suited for explaining the Riemann Hypothesis is that in the context of algebraic quantum groups (e.g. van Daele), as it can be seen in the articles interpreting the Riemann- Mangoldt-Weil exact formula, using (multiplicative) multipliers as the essentioal ingredient for defining distributions (reminiscence of the approach via analysis). In conclusion, instead of a sheaf approach, viweing rationals as functions on the spectrum of the integers (algebraic geometry approach), or taking the set of inequivalent valuations as a starting point (analysis approach), the author prefers the interpretation of the rationals as a function field, with the categorical origin in mind, rather then that of a number field and then geometrizing and categorifying the framework (geometric points and sheaves). The role of a unit, of the odd even prime p = 2 will be explained elsewhere, and in this article it will be treated on equal footing with the other primes,

4 4 LUCIAN M. IONESCU even though the 2 k symmetry/factor of p 1 is clearly a reflection, linked to the orientation of the corresponding cycles in Aut(F p, +). Definition 2.3. The total weight of a prime is W(p) = p p 1 ν p (p 1)). Here ν p (n) = k iff k is the exponent of the prime p in the factorization of n: p k n (ν p is the p-adic valuation). For example W(19) = = 3, since 19 1 = These two functions will have natural interpretations when associating rooted trees to primes, reflecting the complexity of the structure of the corresponding primary fields. As a question to the reader, it would be interesting to investigate the implications of Proth Theorem on the structure of this POSet. The homology of the POSet of primes seems to be related to the structure of the rooted trees. 3. Prime Numbers and Rooted Trees To each prime p we associate a rooted tree t(p) representing the hierarchy of generators of the F p, when applying the internal functor Aut Ab repeatedly. Since the elementary divisors Z p k arise in this way, we have to extend the framework, and extend the mapping between natural numbers, freely generated by primes, and the Hopf algebra of rooted trees, where the elements are linear combinations of forests (families of rooted trees), and concatenation of forests is extended as a bilinear mapping. We define t : N H rt as the multiplicative mapping (m, n) = 1, t(mn) = t(m)t(n), intertwining the Euler phi function and the pruning operator B : t(φ(n)) = B (t(n)). Recall that φ(n) = U(Z n ) is a multiplicative function with values the number of elements of the multiplicative group of Z n, which corresponds to its automorphism group as a Z-module ( discrete space of vectors ). Also B is the adjoint of B + which adjoins a common root to the rooted trees of a forest of H rt. At categorical level, t(p) reflects the primary decomposition of Aut Ab (F p, +), and t is the Grothendieck shadow of the automorphism torsor Aut Ab : Ab Ab Aut iso N φ t H rt iso N t H rt. Natural numbers are viewed here as functions with finite support N = Hom c (P, N), rather then divisors (elements of g P ), consistent with the interpretation of rational numbers as a formal group/ function field Q + = Hom(P,Z), explained below. Explicitly: t(2) =, t(p k ) = B + (t(p k 1, t(φ(p))), and t( l 1 p ki i ) = l 1 B t(p ki i.

5 AN EXPLORATION OF THE PRIME NUMBERS 5 For example, for p = : together with the substitution t(p) = B + (t(φ(p)) = B + (t(2), t(3 2 ), t(5)), t(3 2 ) = B + (t(φ(3) 3)) = B + (t(2), t(3)) = B + (, B + ( )). Note that prime numbers are mapped to rooted trees, which form the algebraic bases of forests of H rt. Once the mapping t from natural numbers to forests is defined, it is natural to study the properties of primes and their distribution within the better understood Hopf algebra of rooted trees. 4. The rational numbers as a formal group The rational numbers, when completed using a p-adic norm, lead to fields of elements which behave like power series (Hensel s analogy). We claim that the field of rational numbers can be interpreted as a formal group. Pursuing this idea and interpret in this way number fields as deformations of function fields, will be investigated elsewhere An external composition on prime numbers. The Euclid s trick (take products of primes, including 2, and add one), almost allows to define a composition of primes. It corresponds to the root joining operator B + on rooted trees (and forests). If p and q are primes, (p 1)(q 1), which corresponds to joining their symmetries at categorical level, might not be the symmetries of a finite field, since p q = (p 1)(q 1) + 1 is not necessarily a prime number. Transport this definition on g P, as a fusion rule 3 : X p X q = r P k(r)x r, (p 1)(q 1) + 1 = p P r kp. and extend it linearly, still denoted. Lemma 4.1. an associative binary operation on g P with identity X 2. Remark 4.1. This leads to a framework reminiscent of star products and deformation quantization: g P plays the role of a Lie (bi)algebra, Q is the associated group, and p-adic numbers Q p = F p ((h)) are a deformation of the laurent series in the formal parameter h. This idea will be explored elsewhere The rationals as a formal group. To make the multiplicative group of rational numbers a formal group, change coordinates, defining the (multiplicative) position vector of x Q by r x = x 1 (difference between x and the neutral element). Then define x y = F(r x, r y ) = r x + r y + r x r y. 3 HomAb is an internal functor.

6 6 LUCIAN M. IONESCU Proposition 4.1. There is a natural isomorphism of formal groups: Exp : g P (Q +, ), Exp(X p) = p 1, where (g P, +) is the additive formal group with addition [14]: F(X p, X q ) = X p + X q, and the multiplicative formal groups (Q +, ). The inverse is called the formal logarithm. Remark 4.2. The exponential implements a tangent map point of view, conform Lie Theory, to reflect the action of the automorphism functor Aut Ab, and the fact that the basic finite fields F p are tangent spaces of the p-adic k-th order truncations of the p-adic quantum (abelian) groups. Proceeding with the investigation in this direction is supported by the presence of theta functions and modular groups in the framework of Riemann Hypothesis, which have deep links with geometric/deformation quantization, and invariants of knots [10] (see also Khovanov s categorification of the invariants of knots [11]) Further avenues of exploration. The presence of this two fundamental God made structure, prime numbers and rooted trees, and the link between them is highly intriguing: why two? Taking as a working hypothesis that they are asymptotically the same, maybe the prime cycles the cohomology of rooted trees: see the universal cohomological problem [12], p.599, we argue that rooted trees as a basis in the Hopf algebra of trees, could be a more fundamental probabilistic space for primes as a random variable : C : (H rt Q, X : (PRT, ρ ) (N, card). where PRT is the subspace in g P generated by prime rooted trees, the image of t : P RT, so that the image of the random variable X is the set of primes P N. The linear mapping C defined on the basis of rooted trees by extending the random variable X from PRT to RT should be a section of t. The problem is to construct it using somehow the contraction where τ : g P g P, τ(x p ) = Div(Exp(X p )), Div(n) = k(p)x p, n = p P p k(p), is the principal divisor of the rational function k : P Z associated to the natural number n (more generally, to any non-zero rational number). Remark 4.3. The author s program is to establish that number fields are function fields, at least for finite extensions of Q, via p-adic numbers, using technique of quantum groups, and exploit this in the direction of better understanding the Riemann Hypothesis (already proved fro function fields by Weil). 5. The Prime Number Theorem The general idea is to reduce the PNT, via linearization, to a simple, Riemann- Roch-like statement, easely prooved at the level of Hopf algebras of rooted trees, its basis.

7 AN EXPLORATION OF THE PRIME NUMBERS 7 The prime Number Theory says that the probability density function of prime numbers is assymptotically 1/ ln n: P(1 < X < n) = π(n)/n 1/ ln(n), n N. Now ln(n) naturally lives in g P, the linearization of Q. Remark 5.1. ln(n) = k(p)ln(p) is a divisor limiting the number of basic elements X p in g p, in a way similar to Riemann-Roch Theorem. This cannot be a simple coincidence. The weight ln(p) present in the various formulas used in the proof of PNT via the Riemann zeta function is in fact a measure. It appears under the disguise of the Mangold function Recall: Λ(p k ) = ln(p), (orλ(n) = 0 otherwise). Ψ(x) = n<xλ(n), The natural measure on g P close to the measure ln(p)dp is the size of the corresponding rooted tree t p, i.e. its degree: Conjecture 5.1. An experimental check is in progress. deg(t p ) = number of nodes. ln(p) deg(t p ) An analog of Riemann-Roch Theorem for Q. In view of the functional interpretation of the rational numbers, the PNT has two parts. PNT refers to the number of delta functions δ p : P N (primes) with respect to the total number of functions k : (P, ln(p)dp) (N, card = dn) with bound exp(k) <= n, where exp is the completely multiplicative function: Note that the composition exp(k) = p P p k (p), {k exp(k) <= n} = n. ln exp : Q + N, k k(p)ln(p) is the integral integral associated to the Mangold measure ln(p) on the measure space (P, ln(p)dp): P k(p)ln(p)dp = p P PNT is equivalent to the asymptotic relation: π(n) Li(n) = k(p) ln(p) = ln(exp(k)). n 2 1/ln(t)dt. Remark 5.2. Abel s summation formula allows to transfer this to (N, dn), while the change of variable, with a slight modification of the formula (replace ln(t) by ln(t) 1), yields an integral of the (e u u)/u function.

8 8 LUCIAN M. IONESCU Recall that the Riemann s inequality part of the Riemann-Roch Theorem [?], says that for a divisor D = p P k px p (matching the above notation when possible), the dimension l(d) of the vector space of meromorphic functions with principal divisor (f) + D >= 0, satisfies: l(d) >= deg(d) g + 1, with equality for large degree divisors (deg(d) = k(p)). Thinking classically P = SpecZ, that primes are points on the discrete line Z (abelain groups are discrete vector spaces ; e.g. DFT etc. [?]), and interpreting k M as a divisor (a homological chain, rather then a finitely supported function): deg(k) = k(p). But we need the ln(p) weight which suggests that there is an underlying fractal manifold structure 4 corresponding to the above POSet structure of P. Then each point p has to be blown-up due to its internal structure (Aut Ab (F p, +)), obtaining the correct degree, accounting not only for multiplicaity, but also for hierarchic structure 5 : deg(k) = k(p)ln(p) = k(p)dm. p P P Then the counting of primes and their density resembles a Riemann-Roch cut-off condition (dimensions of vector spaces equal number of basic elements), where the condition p l < x operates as a ultra-violet cut-off limiting the multiplicity of primes and the primes themselves. What is the analog of the genus g in the RR-formula l(d) >= deg(d) g + 1?. Note also that the second (integer) Chebyshev function is an area determined by the cutt-off k n (p) = [lnn/ ln(p): ψ(n) = ln(p) = n(p)dm, p l P <n while the first Chebyshev function is the corresponding volume of primes θ(n) = ln(p) = dm. p<n supp k n Remark 5.3. One starts to wonder (continues!) whether Minkovski s Lattice Theorem is at work in the lattice exp : g P Q + R, since not only the direction of the inequality in RR-Theorem has to be reversed, but also we are not counting rational numbers ( meromorphic functions with poles at primes), but rather only irreducible polynomials δ p within the polynomial functions k(p) satisfying an integration bound k <= n k(p)dm <= ln(n). 4 Introducing fractal manifolds by gluing p-adic fields viewed as pointed formal manifolds a la Kontsevich, will be attempted elsewhere. 5 Haar analyis point of view.

9 AN EXPLORATION OF THE PRIME NUMBERS Breaking Down The PNT. A general setup for the PNT (distribution of primes) has two levels: at categorical level we have a semi-simple monoidal category (C,, {S p } p P ) with simple objects S p, e.g. finite sets S fin, which induces a partial order on the Grothendieck rig, e.g. the (N, <) (total order here). At the level of iso classes, we have the functions on the simple objects M = (F(P; N), w(p)) corresponding to the objects of the category, together with a weight function w(p) defining the measure space (M, dm), with measure dm = w(p)dp A probabilistic framework. Then the following diagram relates to the divisor interpretation sketched above: χ (M, dm) (N iso,, dn) R ln(n) (R, +, dx) with the exponential isomorphism of groups (not the formal groups): χ( k(p)x p ) = p k(p), k(p)dm = k(p)w(p). P p P For our prototypical example of finite sets C = S fin, the weight function is w(p) = ln(p), dm is the Mangoldt measure 6, and the Mangoldt integral is the formal (e)valuation: k(p)x p k(p) ln(p) = ln(n). p P p P Then we have: n = card{k N k <= n} = card{k M k(p)dm <= lnn, π(n) = card{p p <= n} = card{δ p δ p dm <= lnn, P with d p the unit delta function supported at p (basis dual to the basis {X p } p P of g P ). This allows to translate the equivalent (first Chebyshev) form of the PNT: θ(x) x, θ(x) = ln(p) = dm, p<=x P B(ln x) in terms of functions and measure theory / probability spaces Here B(r) = {k k 1 <= r is the ball of radius r relative to the L 1 -norm (see Remark 5.3). The main point is that a weight function w(p) is expected and that it is essentially the inverse of the (formal) exponential isomorphism χ. We expect to be able to push this approach further and clarify the underlying reason for the so called surprising Prime Number Theorem, if not providing a shorter, algebraic proof. 6 Related to the Mangoldt function Λ(n).

10 10 LUCIAN M. IONESCU 5.4. On Chebyshev s estimates. Another possible application of the POSet of primes and of the functional model of rationals is to the direct count of the primes. This allows to pinch the number of primes, as in Cebyshev s estimates: A 1 x/ lnx π(x) A 2 x/ ln(x). The proof from [9], p.140, aims for determining a lower bound function based on the inequality ( ) 2n < (2n) π(2n), n and an upper bound function derived from a finite difference growth upper bound ( ) 2n n π(2n) π(n) <, n based on a comparison with the same combinatorial coefficient which allows to link the additive world of natural numbers with the multiplicative world of primes. The idea is to interpret the above difference inequality as a finite partial difference π(exp(k(x p ))/ X 2 = π(2 n) π(n) in the direction of the generator p = 2 of the formal group of rationals, and study its growth in analogy to the theory of analytic Lie groups. 6. Conclusions In this article a few ideas are presented, as an alternative approach to the usual directions of study of prime numbers. The approach shifts the emphasis from complex analysis (analytic number theory) to deformation theory as an algebraic modern version of the theories of Lie and Klein [15]. The partial order on the set of primes, derived from the structure of symmetries of basic finite fields, represents a refinement of the total order of primes by size, inherited from the total order on the natural numbers. Second, viewing the multiplicative group of rational numbers as a measure space seems the right setup for a direct, yet conceptual proof of the Prime Number Theorem. Alternatively, a formal group interpretation of the rational numbers, with its Lie graded module of primes as generators, invites to an application of finite PDEs techniques to estimate the growth of the number of primes function π(x). We hope that the reader will be able to see further into these preliminary trials, and possibly find further applications, either to new results, or allowing to simplify some of the old, and usually intricate proofs regarding the distribution of prime numbers. References [1] L. M. Ionescu, Remarks on physics as number theory, gsjournal.net/science- Journals/Essays/View/3617 [2] Yuri Manin, Lectures on zeta functions and motives, MPI / [3] M. J. Shai Haran, The mysteries of the real prime, Oxford University Press, [4] Paula Tretkoff, Noncommutative geometry and number theory, Clay Mathematics Proceedings, Vol.6, [5] I. V. Volovich, Number theory as the ultimate physical theory, [6] Yu. I. Manin, Mathematics and Physics, Birkhauser, Boston, [7] L.M. Ionescu, p-adic numbers and algebraic quantum groups, work in progress.

11 AN EXPLORATION OF THE PRIME NUMBERS 11 [8] L.M. Ionescu, Real numbers and p-adic numbers: a Haar analysis point of view, work in progress. [9] B. Fine and G. Rosenberger, Number Theory: an introduction via the distribution of prime numbers, Birkhauser, [10] Razvan Gelca, Alastair Hamilton, Classical theta functions from a quantum group perspective, [11] M. Khovanov, A categorification of the Jones polynomial, [12] Joseph C. Varilly, Hector Figueroa, Jose M. Gracia-Bondia, Elements of Noncommutative Geometry, Birkhuser Advanced Texts, [13] L.M. Ionescu, from Lie theory to deformation theory and quantization, arxiv.org/abs/ , [14] Formal groups, en.wikipedia.org/wiki/formal group. [15] L. M. Ionescu, From Lie Theory to Deformation Theory and Quantization, arxiv: Department of Mathematics, Illinois State University, IL address: lmiones@ilstu.edu