MATH 566, Final Project Alexandra Tcheng,

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1 MATH 566, Final Project Alexanra Tcheng, The unrestricte partition function pn counts the number of ways a positive integer n can be resse as a sum of positive integers n. For example: p 5, since can be written as + + +, + +, +, +,. This project intens to present a proof of the following remarable result: Theorem: If n, then pn A n n where A n 0 h< ish, in h an sh, r r hr [ ] hr This theorem, prove by Raemacher in 97 [6], greatly improves the asymptotic relation iscovere by both Hary an Ramanujan in 98 [] an J.V.Uspensy in 90 [7] inepenently, an which state that: pn n n e as n where The proof of this formula involves the circle metho, a tool use many times by Hary, Littlewoo an Ramanujan in tacling asymptotic problems of aitive number theory. For example, this metho provies a lot of insight for a possible proof of Golbach s conjecture, which states that every integer is the sum of three primes [5]. Later on, Erős prove that this formula coul be erive by elementary means []. Hary an Ramanujan also prove the following exact asymptotic formula: pn P n + O/ <α n where α is a constant, an P n is the ominant term, asymptotic to n en. ote that the infinite sum + P n iverges for each n. However, Raemacher s choice of contour of integration, which also uses the circle metho, yiels a formula involving a convergent series. The following proof is taen from [], an is ivie into 6 main parts. Proof.. Generating function corresponing to pn: We shall mae use of the following lemma: Lemma: If x <, then n0 pnxn x : F x. Proof. This proof can be foun in []. Restrict x to lie on the interval 0 x < an introuce two functions F m x m x an F x x lim m F mx 5 Since for 0 x <, x converges absolutely, so oes x, an hence F x. Also F m+ x m+ x x m+ m x x m+ F mx F m x 6

2 so that for fixe x, F m x F x for every m. ow, m m F m x x x n + p m x n absolutely convergent since each x n is. n where p m is the number of solutions of the equation m m, ie: the number of partitions of into parts not exceeing m. Since we have p m p, with p m p if m, it hols that lim m p m p. Split F m x into two parts: >0 {}}{ m F m x + p m x p m x + p m x 0 m+ m px F m x F x 7 0 Therefore 0 px converges. Also p m x px F x 8 0 so for each fixe x, p m x converges uniformly in m. Taing lim m : F x lim m F mx lim m 0 0 p m x lim p mx px 9 m This proves the lemma for 0 x <. Tae the analytic continuation to the unit is x <. F x is absolutely convergent on x <. Diviing on both sies of the above by x n+ yiels the Laurent series F x ansion of x, vali for 0 < x < : n+ F x x n+ px x n+ 0 0 This ansion has a simple pole at x 0. Therefore, by Cauchy Resiue Theorem,: F x F x Res x0 pn x xn+ i xn+ C being any contour in 0 < x < enclosing 0. We now move from the x-plane to the τ-plane by performing the following change of variables: x e iτ. The puncture is is mappe to the infinite strip 0 < Rex < in the τ-plane, an in particular, the circle γ : [0, ] D 0 given by γa e +ia is mappe to the straight line joining i to i +. Integrating along γ in this plane gives: pn i γ 0 C 0 F e iτ τ eiτn. Raemacher s path: We now wish to replace the path of integration γ by a path that lies close to the singularities of F x. Since x 0 for x e ih/, where h, with 0 h <, we see that we must tae a path lying near all roots of unity. But before efining it, two efinitions are in orer: Farey fractions: Def: The Farey fractions of orer n are efine as the set of reuce fractions in the close interval [0, ] with enominators n, liste in increasing orer of magnitue. For example: F : 0, F : 0,, F : 0,,,,...

3 For circles: Def: The For circle C h,, where h, with 0 h <, is a circle of raius, with center locate at h + i. ote that C h, is tangent to the real axis at the point x h. The following lemma is given without proof the proof isn t har, but it is lenghty: Lemma: For circles of consecutive Farey fractions are tangent to each other. If h in F, the points of tangency are given by: < h < h are consecutive α h, h + + i + an α h, h + + i + So for example, the five For circles associate to the Farey fractions F { 0,,,, } are tangent to each other, as illustrate on Figure. With these facts at han, it is now possible to efine Raemacher s path P in the following way: Def: Consier the For circles of the Farey series F. If h < h < h are consecutive in F, the points of tangency of C h,, C h, an C h, ivie C h, into two arcs, an upper arc an a lower arc. P is the union of the upper arcs so obtaine. See Figure for an illustration of P. Figure : For the moment we eep fixe, an will let it go to + only near the en of the proof. We may then rewrite the path of integration as: i+ i P 0 h<, h, γh, : h, where γh, is the upper arc of the For circle C h,. This allows us to wor on each arc separately. γh, 5. Changing variables: The following change of variables maps the For circles C h, onto the circle of raius / with center z /: z i τ h τ h + iz 6

4 The points of tangency α h, an α h, of C h, with the neighbouring circles C h, an C h, are mappe to the points z h, an z h, on given by: z h, + + i + an z h, + i + 7 This is illustrate on Figure. Figure : The partition function is now written as: F e iτ pn τ 8 i h, γh, eiτn i inh zh, nz ih h, z h, F z z 9 ote that on, Rez > 0. In orer to perform the estimates in part 5 of the proof, the following facts are useful: Theorem: If z is on the chor joining z h, an z h,, we have z <, an the length of that chor is < /. Proof. We have: z z + 0 Similarly z. If z is on the chor joining z h, an z h,, z max z, z. Using: z < An similarly z <, giving z <. As for the length of the chor, it is z + z <.. Deein s functional equation resse in terms of F :

5 The Deein eta function is of central importance in many applications of elliptic moular functions to number theory. It was introuce by Deein in 877 an is efine in the half-plane H {τ : Imτ > 0} by the equation: ητ e iτ/ n e inτ The functional equation satisfie by ητ is given in the following theorem, given without proof: a b Theorem: If Γ, Γ being the moular group, an c > 0 we have c η [ ] aτ + b a + icτ r hr i + s, c + ητ where sh, cτ + c r In terms of F, this functional equation can be restate as: Theorem: Let F t t an ih x z where Rez > 0, > 0, h, an hh mo. Then ih x z [ ] hr z F x ish, z z F x ote that when z is small, we have ih x z e ih/ ih x z 0 5 So that the behaviour of F near the singularity e ih/ is given by: z F x ish, z z F x 6 Proof. First notice the following relation: Therefore: F e iτ aτ + b η cτ + ητ e iτ e iτ/ e iτ/ eiτ eiτ/ ητ [ a + i c F iτ F η aτ+b cτ+ [ i i aτ + b cτ + 7 ] icτ + s, c + ητ 8 ] a + icτ + s, c + 9 c [ i a + c i τ aτ + b 0 cτ + ] icτ + + s, c Choosing a H, c, h, b hh+ ih F z F Replacing z by z/ yiels the result., an τ iz+h, then τ i z +H ih z z an z z + ish, 5

6 We now have: pn i inh zh, nz h, z h, i inh zh, nz h, z h, ish, h, i 5 inh F ih z z } {{ } x ish, zh, z h, nz z z z z z F x z z F x z Rewriting F x + F x, we can split the integral into two parts: pn i 5 inh zh, nz ish, h, z h, z z z + F x z i 5 inh zh, nz ish, h, z h, z z z z + h, i 5 inh zh, nz ish, z h, z z z F x z : I : I 5. Estimates for I an I : We first estimate I, an start by moving the path of integration from the arc of joining z to z to the chor joining z an z. The integran can then be boune as follows: nz z / z z [ ih F z [ nz z / z z ihm pm m0 rop {}}{ n Rez z / Re z Rez pm ihm m n < n z / z / Re z m <pm {}}{ pm m pm mre z ] m ] z m 5 z mre/z 6 {}}{ m / Re z 7 < n z / pm m / 8 : c z / m where c is a constant that oes not epen on z nor. Since z < / on the chor, an the length of the chor is < /, we have: 9 6

7 I zh, z h, nz h, z z i 5 inh z F x z < c ish, I < C 5 0 h< : C 0 O We now estimate I : here, instea of only integrating over the arc of joining z to z, we integrate over the whole circle. Going clocwise aroun the circle: zh, zh, 0 zh, zh, + + : J + +J J J 0 z h, z h, z h, Estimating J : nz z z z {}}{ n Rez z J Similarly: J C Therefore h, h, h, h, h, zh, 0 < C. n nz i 5 inh ish, I i 5 inh i 5 inh i 5 inh i 5 inh ish, ish, z z : C zh, z h, ish, nz nz nz z h, Re z Rez rop : C z z z h, C z z z z z z ish, J i 5 inh h, z z z z J J z z ish, J an by the above: i 5 inh ish, J + i 5 inh ish, J h, < h, 0 h< 0 h< 5 J + 5 C 0 h< + 5 J 0 h< 5 C O 7

8 Thus: pn h, i 5 inh ish, I + i 5 inh ish, I h, nz 0 h< 0 h< i 5 inh ish, i 5 inh ish, z z z z z z z + O z + O 6. Getting the final answer using Bessel functions: We start by letting, the number of circles use to construct Raemacher s path, go to +. A n 0 h< ish, in h : pn lim + 0 h< + i 5 A n i 5 inh ish, z z z z z z z Defining z + O We then perform two successive changes of variables: First let w z circle to the infinite line Rez : so that z w w. This maps the Seconly, let t w + pn i i i A n z z z z 5 + i or w t 5 A n 5 A n to get: + pn i 5 A n i + i i + i i + i i 5 A n + + oting that if c > 0 an Reν > 0: + i i 5 A n 5 A n i I ν z z/ν i w w w 5 w w w w w 6 w 7 w 5 w w 8 w 5 t t + i t 5 i + i c+ i c i 8 t 5 i t 6 t 6 t t t 9 t 50 t 5 t t ν e t+z /t t 5

9 where I ν z i ν J ν iz, we have that if we let z 6, then: + pn A n i + i 5 A n 6 5 A n + 6 A I / z t 5 i t 6 6 z i + i t t t i + i t t z i i t I / z t 6 t t t But Bessel functions of half-orer can be reuce to elementary functions: I / z z z z z 5 We rewrite everything in terms of n: z 6 z n + z 5 z n z n z n n 55 Also: I / I / z z z z z n n n n So that: pn + A I / z 9

10 as require A n + A n n A n n n References [] Apostol, Tom M., Introuction to Analytic umber Theory. Unergrauate Texts in Mathematics, Springer- Verlag, ew Yor, 976. pp [] Apostol, Tom M., Moular Functions an Dirichlet Series in umber Theory. Grauate Texts in Mathematics, Springer-Verlag, ew Yor, 976. pp.9-0. [] Erős, P. On an elementary proof of some asymptotic formulas in the theory of partitions. Annals Math., :7-50, 9. [] Hary, G.H. an Ramanujan, S. Asymptotic formulae in combinatory analysis. Proc. Lonon Math. Soc 7 98, [5] Miller, S., Taloo-Bighash, R. An Invitation to Moern umber Theory, Princeton University Press, 006. [6] Reemacher, H. On the ansion of the partition function in a series Ann. of Math. 9, 6-. MR 5, 5a. [7] Uspensy, J.V. Asymptotic formulae for numerical functions which occur in the theory of partitions. Bull.Aca.Sci.URSS6 90,