A Note on Goldbach Partitions of Large Even Integers

Transcription

1 arxiv: v3 [mah.nt] Jan 25 A Noe on Goldbach Pariions of Large Even Inegers Ljuben Muafchiev American Universiy in Bulgaria, 27 Blagoevgrad, Bulgaria and Insiue of Mahemaics and Informaics of he Bulgarian Academy of Sciences ljuben@aubg.bg Submied: July 8, 24; Acceped: Jan..., 25 Mahemaics Subjec Classificaion: 5A7, P32, 6C5, 6F5 Absrac Le Σ 2n be he se of all pariions of he even inegers from he inerval (4, 2n], n > 2, ino wo odd prime pars. We show ha Σ 2n 2n 2 /log 2 n as n. We also assume ha a pariion is seleced uniformly a random from he se Σ 2n. Le 2X n (4,2n] be he size of his pariion. We prove a limi heorem which esablishes ha X n /n converges weakly o he maximum of wo random variables which are independen copies of a uniformly disribued random variable in he inerval (,). Our mehod of proof is based on a classical Tauberian heorem due o Hardy, Lilewood and Karamaa. We also show ha he same asympoic approach can be applied o pariions of inegers ino an arbirary and fixed number of odd prime pars. Inroducion and Saemen of he Main Resul For a given sequence of posiive inegers Λ = {λ,λ 2,...}, by a Λ-pariion of he posiive ineger n, we mean a way of wriing i as a sum of posiive

2 inegers from Λ wihou regard o order; he summands are called pars. Le P = {p,p 2,...} be he sequence of all odd primes arranged in increasing order. A prime pariion is a Λ-pariion wih Λ = P. Le Q(n) be he number of prime pariions of n. Hardy and Ramanujan[6,7] were apparenly he firs who sudied he asympoic behavior of he number of ineger (Λ = {, 2,...}) and prime pariions for large n. For prime pariions hey proved he following asympoic formula: n logq(n) 2π 3logn, n. The sudy of he asympoic behavior of Q(n) iself is quie complicaed. I urns ou ha he corresponding asympoic formula conains ranscendenal sumsoverheprimeswhichcanbeexpressed inermsofzerosofheriemann zea funcion (for more deails, see e.g. [9; p. 24]). Recenly Vaughan [6] proposed and sudied a modificaion of he problem, where n is replaced by a coninuous real variable. His asympoic resuls avoid ranscendenal sums over primes. Consider now he number Q m (n) of prime pariions of n ino m pars ( m n). The bivariae generaing funcion of he numbers Q m (n) is of Euler s ype, namely, G(x,z) = + n z n Q m (n)x m = ( xz p k ) () n= m= p k P (he proof may be found in [; Secion 2.]). In his noe we focus on he asympoic behavior of he coefficiens Q 2 (n) of x 2 and z n in he power series expansion of G(x,z) in powers of x and z. For n > 4, Q 2 (n) couns he number of ways of represening n as a sum of wo odd primes. Obviously, Q 2 (n) = ifnisodd. In742GoldbachconjecuredhaQ 2 (n) forevery even ineger n > 4. This problem remains sill unsolved (for more deails, see e.g. [8; Secion 2.8 and p. 594]). Anoher famous conjecure relaed o prime pariions was saed by Hardy and Lilewood [5], who prediced he 2

3 asympoic form of Q 2 (n) for large even n. They conjecured ha Q 2 (n) 2C 2 p k n du p k 2 p k P,p k n 2 log 2 u 2C 2 p k n p k 2 log 2, n, (2) n p k P p k P,p k n where C 2 is he win prime consan C 2 := ( ) (p k ) 2 = (for he role of C 2 in he disribuion of he prime numbers, see again [8; Secion 22.2]). This conjecure remains also sill open. In he presen noe we do no deal wih he asympoic equivalence (2) bu consider he sum funcion S(2n) = Q 2 (2k), n > 2, (3) 2<k n couning all pariions of he even inegers from he inerval (4, 2n] ino wo odd prime pars. Someimes his kind of pariions are called Goldbach pariions. Le Σ 2n denoe he se of hese pariions. Our main resul is he following asympoic equivalence. Theorem We have Σ 2n = S(2n) 2n2 log 2 n, n. Consider now a random experimen. Suppose ha we selec a pariion uniformly a random from he se Σ 2n, i.e. we assign he probabiliy /S(2n) o each Goldbach pariion. We denoe by P he uniform probabiliy measure on Σ 2n. Le 2X n (4,2n] be he number ha is pariioned by his random selecion. 2X n is also called he size of his pariion. Using Theorem, we deermine he limiing disribuion of he random variable X n. 3

4 Theorem 2 If < u <, hen lim n P ( ) Xn n u = u 2. Remark. Using he Prime Number Theorem [8; Secion.8], i is easy o show ha he number of ordered pairs of primes no exceeding 2n is also 2n 2 /log 2 n; cf. wih he resul of Theorem. Hence, we conclude ha almos all even inegers ha are 2n have only one pariion ino wo prime pars. Remark 2. In probabilisic erms Theorem 2 shows ha he ypical size of a random Goldbach pariion is a fracion of 2n. Moreover, Theorem 2 implies ha X n /n converges weakly, as n, o a random variable whose cumulaive disribuion funcion is F(u) = if u, u 2 if < u <, if u. I can be easily seen ha F(u) is he disribuion funcion of max{u,u 2 }, where U and U 2 are wo independen copies of a uniformly disribued random variable in he inerval (,). Remark 3. One reason o sudy he sum funcion (3) is moivaed by a resul due o Brigham [2]. He has sudied he asympoic behavior of a similar sum funcion relaed o ineger pariions weighed by he sequence of he von Mangold funcions (he definiion of a von Mangold funcion and is role in he proof of he Prime Number Theorem may be found in [8; Secion 7.7]). The asympoic behavior of a single erm in Brigham s sum funcion was subsequenly sudied by Richmond [3] and Yang [7]. Their resuls are essenially based on Brigham s observaions. Remark 4. Anoher ineresing problem on prime pariions is relaed o he asympoic behavior of he coefficiens Q m (n), he number of prime pariions of n wih m pars (see ()). Haselgrove and Temperley [9; p. 24] found an asympoic form for Q m (n), whenever m = m(n) as n in a proper way. In probabilisic erms heir resul can be saed as follows. Consider a random variable, whose probabiliy disribuion funcion is defined by he raio Q m (n), m =,...,n. (4) Q(n) 4

5 Haselgrove and Temperley [9] showed ha his random variable converges weakly o a non-degenerae random variable as n. They also deermined he momen generaing funcion of his limiing variable. The asympoic form of he mean and he variance of probabiliy disribuion (4) were found recenly by Ralaivaosaona [2]. Our paper is organized as follows. Secion 2 conains some preliminaries. TheproofsofTheoremsand2aregiveninSecion3. Ourmehodofproofis essenially based on a classical Tauberian heorem due o Hardy, Lilewood and Karamaa (see [4]). Finally, in Secion 4 we presen an exension of our main resul. In paricular, we show ha he same approach yields similar resuls for prime pariions of n ino m > 2 pars whenever m is fixed ineger. 2 Preliminary Resuls We sar wih a generaing funcion ideniy for he sequence {Q 2 (2k)} k>2 of he couns of Goldbach pariions. Lemma For any real variable z wih z <, le f(z) = p k Pz p k. (5) Then, we have 2 k>2q 2 (2k)z 2k = f 2 (z)+f(z 2 ). (6) Proof. Differeniaing he lef-hand side of () wice wih respec o x and seing hen x = and m = 2, we ge 2 G(x,z) n x 2 x=,m=2 = z n m(m )Q m (n)x m 2 x=,m=2 n= m=2 = 2 2 (n)z n=q n = 2 Q 2 (2k)z 2k. k>2 ThelasequaliyfollowsfromheobviousideniiesQ 2 () = Q 2 (2) = Q 2 (4) = and Q 2 (2k +) = for k =,2,... The righ-hand side of () can be also 5

6 wrien as exp( p k P log( xzp k )). Differeniaing i wice, in he same way we find ha ( ( 2 G(x,z) x 2 x= = exp ))( ) 2 log( xz p k z p k ) xz p x= k p k P p k P ( ( + exp ))( ) log( xz p k z 2p k ) ( xz p k p k P ) 2 x= = f 2 (z)+f(z 2 ), p k P which complees he proof. Furher, we will use a Tauberian heorem by Hardy-Lilewood-Karamaa whose proof may be found in [4; Chaper 7]. We use i in he form given by Odlyzko [; Secion 8.2]. Hardy-Lilewood-Karamaa Theorem. (See [; Theorem 8.7, p. 225].) Suppose ha a k for all k, and ha g(x) = a k x k k= converges for x < r. If here is a ρ > and a funcion L() ha varies slowly a infiniy such ha ( ) g(x) (r x) ρ L, x r, (7) r x hen n ( n a k r k r k= ) ρ L(n), n. (8) Γ(ρ+) Remark. A funcion L() varies slowly a infiniy if, for every u >, L(u) L() as. 3 Proof of he Main Resul Proof of Theorem. We need o show ha power series (6) saisfies he condiions of Hardy-Lilewood-Karamaa heorem. The nex lemma esablishes an asympoic equivalence of f(z) as z. 6

7 Lemma 2 Le f(z) be he power series defined by (5). Then, as z, f(z) ( )( ). log z loglog z Proof. As usual, by π(y) we denoe he number of primes which do no exceed he posiive real number y. In (5) we se z = e, >, and apply an argumen similar o ha given by Song [5] (see also [3]). We have where f(e ) = e y dπ(y) = e y π(y)dy = π(s/)e s ds = I ()+I 2 (), (9) /2 I () = π(s/)e s ds, I 2 () = π(s/)e s ds. /2 For I () we use he bound π(s/) s/. Hence, for enough small >, we obain I () ( = /2 se s /2 + se s ds /2 e s ds ) = O(/2 ) = O( /2 ). () The esimae for I 2 () follows fromhe Prime Number Theorem wih anerror erm given in a suiable form. So, i is known ha, for y >, π(y) = y ( ) y logy +O log 2 y (see e.g. [; Theorem 23, p. 65]). Furhermore, for s /2, we have logs log. Hence, as in [5], we ge 2 ( ) π(s/) = s s log +O +logs ( log +logs) 2 = s ( ( )) ( ) logs s log +O log +O log 2 ( ) = s s(+ logs ) log +O log 2. () 7

8 We also recall ha in () we have used he obvious esimae /2 se s ds = O( /2 ). (2) Combining () and (2), we obain I 2 () = ( ) log se s ds+o /2 log 2 s(+ logs )e s ds /2 = ( ) ( ) log se s ds+o( /2 ) +O log 2 = ( ) ( ) log +O /2 log +O log 2 log, +. (3) Hence, by (9), () and (3), f(e ) log, +. The proof is now compleed afer he subsiuion = log z. Since log z = logz = log( ( z)) z, z, he asympoic equivalence in Lemma 2 becomes Therefore, f(z) f 2 (z)+f(z 2 ) ( z)log, z. z ( z) 2 log 2, z, z which implies ha he series k>2 Q(2k)z2k saisfies condiion (7) of Hardy- Lilewood-Karamaa Tauberian heorem wih r =,ρ = 2 and L() = log 2 8

9 (see also (6)). The asympoic equivalence of Theorem follows immediaely from (8). Proof of Theorem 2. Recall ha 2X n (4,2n] equals he size of a Goldbachpariionhaischosen uniformly arandomfromhese Σ 2n ofallsuch pariions. Since S(2n) = Σ 2n and, for any N (2,n], S(2N) = Σ [2N] ([a] denoes he ineger par of he real number a), from (3) i follows ha P(2X n 2N) = S(2N) S(2n). (4) Seing N un, < u <, andapplying Theorem wice - ohenumeraor and he denominaor of (4), we see ha he limi of (4), as n, is u 2. This complees he proof. 4 Prime Pariions wih More Than Two Pars Le m > 2 be an ineger and le Σ m,n denoe he se of prime pariions of he inegers from he inerval (4,n] ino m pars. The goal of his secion is o exend he resuls of Theorems and 2 o prime pariions from he class Σ m,n. We sae hem below as wo separae heorems. Theorem 3 For any fixed ineger m > 2, we have Σ m,n ( ) m n, n. m! logn Furhermore, le X m,n denoe he size of a prime pariion seleced uniformly a random from he class Σ m,n. (The uniform probabiliy measure on Σ m,n is again denoed by P.) Theorem 4 If < u < and m is as in Theorem 3, hen ( ) lim P Xm,n n n u = u m. Theorem 4 shows a weak convergence similar o ha esablished in Theorem 2. Namely, for any fixed ineger m, X m,n /n converges, as n, o max{u,...,u m }, where U,...,U m are independen copies of a random variable ha is uniformly disribued in he inerval (, ). 9

10 Below we only skech he proof of Theorem 3. The proof of Theorem 4 is almos idenical wih ha of Theorem 2. Skech of he proof of Theorem 3. Our main ool is again he generaing funcion ideniy (). We noice firs ha he coefficiens Q m (n) are = if eiher m is odd and n is even or m is even and n is odd. By he definiion of Q m (n), we also have Σ m,n = k nq m (k). (5) We compue he mh derivaive of he infinie produc in () using Faa di Bruno formula for derivaives of compound funcions (see e.g. [4; Secion 2.8]). We inroduce he following auxiliary noaions: b(x) = b(x,z) := p k P log( xz p k ), b j = b j (x,z) := j b(x,z) x j,j =,...,m. (6) Using formulae (43) and (46) of [4; Secion 2.8], we obain where d m dx meb(x) = e b(x) b m +R m, (7) R m = R m (x,z) = e b(x,z) m! k!...k m! ( ) k b...! ( ) km bm (8) m! and denoes he sum over all inegers k j,j =,...,m, such ha m j= jk j = m and k < m. Seing x = in (6), we find ha b(,z) = and b j (,z) = f(z j ),j =,...,m, where he funcion f(z) is defined (5). Moreover, in he righ-hand side of (8) we have k k m m. In fac, since k < m by he definiion of a leas one k j is > for j 2. Hence if m = k + k k m, hen m < k + m j=2 jk j = m. Since f(z j ) = O(f(z)) as z and since k +...+k m m, we conclude ha R m (,z) = O(f m (z)). Therefore (7) becomes or, equivalenly, d m dx meb(x) = f m (z)+o(f m (z)), m G(x,z) x m x= = f m (z)+o(f m (z)) (9)

11 as z. On he oher hand, m G(x,z) x m x= = m! k m Q m (k)z k. (2) Applying Lemma 2, as in he proof of Theorem, we obain he asympoic equivalence f m (z) ( z) m log m, z. (2) z The observaions in (9)-(2) imply ha m G(x,z) x m x= ( z) m log m, z. z So, condiion (7) of Hardy-Lilewood-Karamaa heorem is saisfied wih r =,ρ = m and L() =. The required resul follows a once from (5) log m and (8). Acknowledgemens I am graeful o he referee for carefully reading he paper and for his helpful commens and suggesions. References [] G.E. Andrews. The Theory of Pariions. Encyclopedia Mah. Appl., no. 2. Addison-Wesley, Reading, MA, 976. [2] N.A. Brigham. On a cerain weighed pariion funcion. Proc. Amer. Mah. Soc., :92-24, 95. [3] P. Erdős and P. Turán. On some problems of a saisical group heory, IV. Aca Mah. Acad. Sci. Hungar., 9:43-435, 968. [4] G.H. Hardy. Divergen Series. Oxford Univ. Press, Oxford, 949. [5] G.H. Hardy and J.E. Lilewood. Some problems of Pariio Numerorum, III: On he expression of a number as a sum of primes. Aca Mah., 44:-7, 922.

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