6 0 On a General Theorem of Number Theory 0 2 Leading to the Gibbs, Bose–Einstein, and n a J Pareto Distributions as well as to the 2 Zipf–Mandelbrot Law for the Stock Market ] h p - c V. P. Maslov o ∗ s . s c i s Abstract y h p The notion of density of a finite set is introduced. We prove a [ general theorem of set theory which refines the Gibbs, Bose–Einstein, 1 and Pareto distributions as well as the Zipf law. v 5 Suppose that (n) is a sequence of finite sets tending as n to an 0 M → ∞ 0 infiniteset. Supposethat ( (n))isthenumberofelementsintheset (n). 1 N M M Theset (n) issaidtobeρ( )-measurable ifthereexists asmoothconvex 0 M · 6 function ρ( ), called a density function, such that the limit 0 · / s ρ( ( (n))) c lim N M i ρ(n) s n→∞ y h is finite. This limit is called the ρ( )-density of the sequence (n) of sets. p · M Let us present a few examples. : v i X Example 1 Consider the eigenvalues of the k-dimensional oscillator with r potential a k U(x) = (ω x2), x Rk, i i ∈ i=1 X ∗ Moscow Institute of Electronics and Mathematics, [email protected] 1 where the ω are commensurable: i ∆Ψ +U(x)Ψ = λ Ψ , Ψ (x) L (Rk). i i i i i 2 − ∈ Suppose that N (λ ) is the number of its eigenvalues not exceeding a given λ i positive number λ. If λ , then the limit → ∞ lnN (λ ) λ i lim = k λ→∞ lnλ coincides with the dimension of the oscillator. Example 2 Suppose that F is a compact set and N (ǫ) is the minimal num- F ber of sets of diameter at most ǫ needed to cover F. Then N (ǫ) is ln( )- F · measurable and its density coincides with the metric order of the compact set F (see [1]). Consider the set of nonnegative numbers λ ,λ ,...,λ and the set 1 1 2 n {M } of integers 1,2,...,N; N = N(n). Suppose that the set is 2 2 {M } {M } ln( )@-commensurable and s is its density: · lnN lim = s. lnn n→∞ Besides, let λ(n) be the arithmetic mean of the ensemble of λ : i n 1 λ(n) = λ . i n 1 X Suppose that we are given a number E(n). Consider the following cases: (1) ǫ E(n) λ(n)N, ǫ > 0; ≤ ≤ (2) E(n) λ(n)N. ≥ Consider the set of mappings of onto . Two mappings are 2 1 {M } {M } said to be equivalent if their images are identical. Further, we shall only consider nonequivalent mappings and denote them by . 3 {M } Suppose that the sum of elements in is equal to N = n N and {M2} i=1 i the bilinear form of the pair of sets and satisfies the condition 1 2 {M } {M } P n N λ E(n) i i ≤ (cid:12)i=1 (cid:12) (cid:12)X (cid:12) (cid:12) (cid:12) (cid:12) 2(cid:12) in case (1) and n N λ E(n) i i ≥ (cid:12)i=1 (cid:12) (cid:12)X (cid:12) in case (2). (cid:12) (cid:12) (cid:12) (cid:12) Note that the set is lnln( )-measurable. 3 {M } · Without loss of generality, we assume that the realnumbers λ ,λ ,...,λ 1 2 n are naturally ordered, i.e., 0 λ λ , and we split the interval 1,2,...,n i i+1 ≤ ≤ into k intervals (to within 1), where k is independent of n: 1,2,...,n , n +1,n +2...,n , n +1,n +2,...,n , ..., 1 1 1 2 2 2 3 k n +1,n +2,...,n , n = n; k−1 k−1 k l l=1 X here l = 1,...,k is the number of the interval. Denotebyλ ,l = 1,2,...,k,thenonlinearaverageofλ overeachinterval: l i nl λ = Φ ψ (λ ) , l αβ αβ i (cid:18)nXl−1 (cid:19) where ψ (x) is the two-parameter family of functions and Φ is its inverse: αβ αβ Φ (ψ (x)) = 1; namely, αβ αβ (a) ψ = αe−βx for s > 1; (1) αβ 1 (b) ψ = for s = 1; αβ αeβx 1 − 1 (c) ψ = for 0 < s < 1. αβ βx+lnα The parameters α and β are related to N(n) and E(n) by the conditions n n ψ (λ ) = N(n), λ ψ (λ ) = E(n). (2) αβ i i αβ i i=1 i=1 X X Consider the subset : 3 A ⊂ M k nl = N ψ (λ ) ∆ , (3) i αβ l A − ≤ (cid:26)Xl=1(cid:18)nXl−1 (cid:19) (cid:27) 3 where √N ln1/2+ǫN for N n, ≪ √nln1/2+ǫn for N n, ∆ =  (4) ∼  N  ln1/2+ǫn for N n √n ≫  is called the resolving power.  Theorem 1 The following inequality holds: ( ) C C 3 N M \A + , ( ) ≤ nk Nk 3 N M where k is arbitrary and C is a constant independent of n and N. Proof. Obviously, n N{M3\A} = Θ N(n)E(n)− Niλi δ ni=1Ni,N(n) (5) {XNi}(cid:18) (cid:26) Xi=1 (cid:27) P k nl Θ N ψ (λ ) ∆ . i αβ l × − − {XNi}(cid:18) (cid:26)(cid:12)(cid:12)Xl=1(cid:18)i=Xnl−1 (cid:19)(cid:12)(cid:12) (cid:27)(cid:19) (cid:12) (cid:12) Here the sum is taken over all inte(cid:12)gers N , Θ(x) is the Heavi(cid:12)side function, i 1 for x 0 Θ(x) = ≥ (0 for x < 0, and δ is the Kronecker delta, k1,k2 1 for k = k , 1 2 δ = k1,k2 (0 for k1 = k2. 6 Let us use the integral representations e−νN π δN,N′ = dφe−iNφeνN′eiN′φ, (6) 2π Z−π 1 ∞ 1 Θ(y) = dx eβy(1+ix). (7) 2πi x i Z−∞ − 4 We have ∞ n ∞ e−β ni=1Niλi dEΘ E N λ e−βE = dEe−βE = . (8) − i i Pβ Z0 (cid:18) i=1 (cid:19) Z ni=1Niλi X P Denote k Z(β,N) = e−β ni=1Niλi, ζ(ν,β) = ζ (ν,β), l P {XNi} Yl=1 nl 1 ζ (ν,β) = ξ (ν,β), ξ (ν,β) = , i = 1,...,n, l i i 1 eν−βλi i=Ynl−1 − and Γ(E,N) = . 3 N{M } Since (E) < (E +ǫ) for ǫ > 0, we have 3 3 N{M } N{M } ∞ Z(β,N) β dE′Γ(E′,N)e−βE′ = Γ(E,N)e−βE. (9) ≥ ZE Therefore, Z(β,N)eβE. (10) 3 N{M } ≤ But, by (6), e−νN π Z(β,N) = dαe−iNαζ(β,ν +iα); (11) 2π Z−π hence (12) 3 N{M \A} e−νN+βE π n exp( iNφ) exp ( β N λ )+(iφ+ν)N dφ j j j ≤ 2π − − (cid:12)(cid:12) Z−π(cid:20) {XNj}(cid:18) (cid:26) Xj=1 (cid:27)(cid:21) (cid:12)(cid:12) k nl Θ N ψ (λ ) ∆ , j αβ l × − − (cid:26)(cid:12)(cid:12)Xl=1(cid:18)j=Xnl−1 (cid:19)(cid:12)(cid:12) (cid:27)(cid:19)(cid:12)(cid:12) (cid:12) (cid:12) (cid:12) where β andν(cid:12)= lnαarerealparamet(cid:12)ers forwh(cid:12)ich theseries isconvergent. − 5 Estimatingtheright-handside, carryingthemodulusthroughtheintegral sign and then through the sign of the sum, and integrating over φ, we obtain n n e−νN expβE(n) exp β N λ +νN (13) N{M3 \A} ≤ − i i i {XNi} (cid:26) Xi=1 (cid:27) k nl Θ N ψ (λ ) ∆ . i αβ l × − − (cid:26)(cid:12)(cid:12)Xl=1(cid:18)i=Xnl−1 (cid:19)(cid:12)(cid:12) (cid:27) (cid:12) (cid:12) Let us use the following inequ(cid:12)ality for the hyperbolic co(cid:12)sine cosh(x) = (ex+ e−x)/2: k k cosh(x ) 2−keδ for all x , x δ 0. (14) l l l ≥ | | ≥ ≥ l=1 l=1 Y X Hence, for all positive c and ∆, we have the inequality (cf˙ [2, 3]) k nl k nl Θ N ψ (λ ) ∆ 2ke−c∆ cosh c N cψ (λ ) . i αβ l i αβ l − − ≤ − (cid:26)(cid:12)(cid:12)Xl=1(cid:18)i=Xnl−1 (cid:19)(cid:12)(cid:12) (cid:27) Yl=1 (cid:18) i=Xnl−1 (cid:19) (cid:12) (cid:12) (15) (cid:12) (cid:12) We obtain 2ke−c∆exp(βE(n) νN) (16) 3 N{M \A} ≤ − n k nl exp β N λ +νN cosh cN cψ (λ ) i i i i αβ l × − − {XNi} (cid:26) Xi=1 (cid:27)Yl=1 (cid:18)i=Xnl−1 (cid:19) = eβE(n)e−νNe−c∆ k ζ (ν +c,β)exp( cψ (λ ))+ζ (ν c,β)exp(cψ (λ )) . l αβ l l αβ l × − − l=1 Y(cid:0) (cid:1) Let us apply Taylor’s formula to ζ (ν + c,β). Namely, there exists a γ < 1 l such that c2 ln(ζ (ν +c,β)) = lnζ (ν,β)+c(lnζ )′(ν,β)+ (lnζ )′′(ν +γc,β). l l l ν 2 l ν Obviously, ∂ lnζ ψ (λ ). l α,β l ∂ν ≡ 6 ′′ Let c = ∆/D(ν,β), where D(ν,β) = (lnζ) (ν,β). The right-hand side of ν relation (16) is equal to k ∆2 ∆2D(ν +γ∆/D(ν,β),β) 2keβE(n)e−νN ζ (ν,β)exp + . l −D(ν,β) 2(D(ν,β))2 l=1 (cid:26) (cid:27) Y Imposing the following constraint on ∆: ∆ D ν + ,β (2 ǫ)D(ν,β), D(ν,β) ≤ − (cid:18) (cid:19) where ǫ > 0, and taking into account the fact that D(ν,β) is monotone increasing in ν, we finally obtain ( ) 2keβE(n)e−νNζ(ν,β)e−ǫ∆2/D(ν,β). 3 N M \A ≤ Next, let us estimate ζ(ν,β). The following lower bound for Z(β,N) was obtained in [2], relation (95): ζ(ν′,β) 27D(ν′,β)Z(β,N)eν′N, (17) ≤ where ν′ = ν′(β,N) is determipned from the condition n ξ (ν,β) = N. (18) i i=1 X Suppose that β = β′ is determined from the condition n λ ξ (ν,β) = E(n). (19) i i i=1 X Since Z(β,N) is determined by the integral (11), its asymptotics given by the saddle-point method (the stationary phase due to Laplace) yields a unique saddle point for α = 0. The square root of the second derivatives with respect to α will appear in the denominator. As a result, we obtain ζ(ν′,β′) CeNν′e−β′E(µ)D(ν′,β′) , 3 ≤ N{M } 7 where C is a constant. Therefore, we finally obtain 1 N{M3 \A} 2kCD(ν′,β′)e−ǫ∆2/D(ν′,β′). (20) ≤ 3 N{M } Further, it is easy to estimate D(ν,β) as a function of N,n: D N for ∼ N < n, while, for n N, the estimate for D yields the relation D N2/n. ≫ ∼ Hence we obtain the estimate for ( )/ , given in the theorem. 3 3 N M \A N{M } Example 3 For the case in which s > 0 is sufficiently small (and hence, k N = N not very large), the Bose–Einstein distribution is of the form l=1 l P N nl e−λiβ l i=nl−1 , (21) Nl+1 ∼ Pni=lnl−1e−λiβ P where β = 1/(kT), T is the temperature, and k is the Boltzmann constant. In the case of a Bose gas, for s < 1, we have a distribution of Gibbs type, i.e., the ratio of the number of particles on the lth interval to the number of particles on the (l+1)th interval obeys formula (21). Example 4 Inthe cases > 1, weobtain a refinementof the Zipf–Mandelbrot law [5], namely, N nl 1 l . (22) n ∼ λ +ν (cid:26)i=Xnl−1 i (cid:27) However, if s is close to 1, then it is better to use relation (b) in (1), which uniformly passes into relation (c) and relation (a). Note that if all the λ on the lth interval are identical and equal to λ(l), i then N /n 1/λ(l), and since n N1/s, it follows that, in this case, we l l ∼ l ∼ l obtain the Zipf–Mandelbrot formula. Example 5 (relation between the sales volume and the prices on the stock market) Let us now consider the relation between the prices and the number of sold (bought) shares of some particular company on the stock market. Since the number n of sold shares of that company during the ith day is i equal to the number of bought shares and λ is the price of the shares at the i 1Alowerboundfor 3 wasobtainedbyG.V.Koval’andtheauthorin[4]without N{M } recourse to the saddle-point method. 8 end of the day, we set, averaging over n days, the nonlinear average price l as nl+1 l λ = Φ φ(λ ) , i (cid:18)iX=nl (cid:19) where φ(x) = 1/(x+ν), ν = const, and Φ(x) is the function inverse to φ(x). Then, by Theorem 1, we have l n Aφ(λ ), (23) l ≃ i where A is a constant. Thus, the stock market obeys the refined Zipf–Mandelbrot law if all the types of transactions are equiprobable (see [6, 7]). In conclusion, note that although the theorem is stated in terms of set theory owing to the fact that we have introduced the notion of equivalent mappings, it belongs, most likely, to number theory. Under the same con- (n) (n) ditions, considering the set of mappings of the set onto the set M1 M2 without the condition for the equivalence of mappings, i.e., considering all mappings, we can obtain a similar theorem that will only be relevant to the refinement of the Gibbs distribution. At the same time, such a theorem is related to information theory and a generalization of Shannon’s entropy. Here the estimate has special features, and the corresponding article will be published jointly with G. V. Koval’. References [1] W. Hurewicz and H. Wallman, Dimension Theory, Princeton Mathe- matical Series, vol. 4. Princeton University Press, Princeton, NJ, 1941; Russian translation: Moscow, 1948. [2] V. P. Maslov, “Nonlinear averages in economics,” Mat. Zametki [Math. Notes], 78 (2005), no. 3, 377–395. [3] V. P. Maslov, “The law of large deviations in number theory. Com- putablefunctionsofseveralargumentsanddecoding,” Dokl. Ross. Akad. Nauk [Russian Acad. Sci. Dokl. Math.], 404 (2005), no. 6, 731–736. [4] G. V. Koval’ and V. P. Maslov, “On estimates for a large partition function,” (to appear). 9 [5] B. Mandelbrot, Structure formelle des textes et communication, Word, vol. 10. no. 1, New York, 1954. [6] V. P. Maslov, “The principle of increasing complexity of portfolio for- mation on the stock exchange,” Dokl. Ross. Akad. Nauk [Russian Acad. Sci. Dokl. Math.], 404 (2005), no. 4, 446–450. [7] V. P. Maslov, “A refinement of the Zipf law for frequency dictionaries andstockexchanges,” Dokl. Ross. Akad. Nauk [Russian Acad. Sci. Dokl. Math.], 405 (2005), no. 5. 10