Rademacher-Type Formulas for Partitions and Overpartitions

Transcription

1 Georgia Southern University Digital Southern Mathematical Sciences Faculty Publications Mathematical Sciences, Department of 2010 Rademacher-Type Formulas for Partitions and Overpartitions Andrew Sills Georgia Southern University, Follow this and additional wors at: Part of the Mathematics Commons Recommended Citation Sills, Andrew "Rademacher-Type Formulas for Partitions and Overpartitions." International Journal of Mathematics and Mathematical Sciences, 2010: doi: /2010/ This article is brought to you for free and open access by the Mathematical Sciences, Department of at Digital Southern. It has been accepted for inclusion in Mathematical Sciences Faculty Publications by an authorized administrator of Digital Southern. For more information, please contact

2 International Journal of Mathematics and Mathematical Sciences Volume 2010, Article ID , 21 pages doi: /2010/ Research Article A Rademacher Type Formula for Partitions and Overpartitions Andrew V. Sills Department of Mathematical Sciences, Georgia Southern University, Statesboro, GA, , USA Correspondence should be addressed to Andrew V. Sills, asills@georgiasouthern.edu Received 3 July 2009; Revised 21 January 2010; Accepted 28 February 2010 Academic Editor: Stéphane Louboutin Copyright q 2010 Andrew V. Sills. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original wor is properly cited. A Rademacher-type convergent series formula which generalizes the Hardy-Ramanujan- Rademacher formula for the number of partitions of n and the Zucerman formula for the Fourier coefficients of ϑ 4 0 τ 1 is presented. 1. Bacground 1.1. Partitions A partition of an integer n is a representation of n as a sum of positive integers, where the order of the summands called parts is considered irrelevant. It is customary to write the parts in nonincreasing order. For example, there are three partitions of the integer 3, namely, 3, 2 1, and Let pn denote the number of partitions of n, with the convention that p0 1, and let fx denote the generating function of pn,thatis,let fx : pnx n. n0 1.1 Euler 1 was the first to systematically study partitions. He showed that fx 1 1 x m. 1.2

3 2 International Journal of Mathematics and Mathematical Sciences Euler also showed that 1 fx 1 n x n3n 1/2, n 1.3 and since the onents appearing on the right side of 1.3 are the pentagonal numbers, 1.3 is often called Euler s pentagonal number theorem. Although Euler s results can all be treated from the point of view of formal power series, the series and infinite products above and indeed all the series and infinite products mentioned in this paper converge absolutely when x < 1, which is important for analytic study of these series and products. Hardy and Ramanujan were the first to study pn analytically and produced an incredibly accurate asymptotic formula 2, page 85, equation 1.74, namely, α n 1 pn 2 ωh, e 2ihn/ d 2 dn 1 O n 1/4), 0 h< h,1 / ) 2/3n 1/24 n 1/ where 1 r ωh, i r1 hr hr 1 2) ). 1.5 α is an arbitrary constant, and here and throughout h, is an abbreviation for gcdh,. Later Rademacher 3 improved upon 1.4 by finding the following convergent series representation for pn: pn 1 ωh, e 2inh/ d 2 dn 1 0 h< h,1 sinh / ) 2/3n 1/24 n 1/ Rademacher s method was used extensively by many practitioners, including Grosswald 4, 5, Haberzetle 6, Hagis 7 15, Hua 16, Isei 17 19, Lehner 20, Livingood 21, Niven22, and Subramanyasastri 23 to study various restricted partitions functions. Recently, Bringmann and Ono 24 have given exact formulas for the coefficients of all harmonic Maass forms of weight 1/2. The generating functions considered herein are wealy holomorphic modular forms of weight 1/2, and thus they are harmonic Maass forms of weight 1/2. Accordingly, the results of this present paper could be derived from the general theorem in 24. However, here we opt to derive the results via classical method of Rademacher.

4 International Journal of Mathematics and Mathematical Sciences Overpartitions Overpartitions were introduced by Corteel and Lovejoy in 25 and have been studied extensively by them and others including Bringmann, Chen, Fu, Goh, Hirschhorn, Hitczeno, Lascoux, Mahlburg, Robbins, Rødseth, Sellers, Yee, and Zhao An overpartition of n is a representation of n as a sum of positive integers with summands in nonincreasing order, where the last occurrence of a given summand may or may not be overlined. Thus the eight overpartitions of 3 are 3, 3, 21, 21, 21, 21, 111, Let pn denote the number of overpartitions of n and let fx denote the generating function n0 pnxn of pn. Elementary techniques are sufficient to show that fx 1 x m fx2 1 xm fx Note that 1 fx 1 n x n2. n 1.8 via an identity of Gauss see equation , page 23 in 44, 45, so that the reciprocal of the generating function for overpartitions is a series wherein a coefficient is nonzero if and only if the onent of x is a perfect square, just as the reciprocal of the generating function for partitions is a series wherein a coefficient is nonzero if and only if the onent of x is a pentagonal number. Hardy and Ramanujan, writing more than 80 years before the coining of the term overpartition, stated 2, page that the function which we are calling pn has no very simple arithmetical interpretation; but the series is none the less, as the direct reciprocal of a simple ϑ-function, of particular interest. They went on to state that pn 1 d e ) n dn n 2 cos 3 n 1 ) d ) 6 e n/3 O n 1/4). dn 1.9 In fact, 1.9 was improved to the following Rademacher-type convergent series by Zucerman 46, page 321, equation 8.53: pn h< h,1 ωh, 2 d ) sinh n/ ω2h, e 2inh/ ). dn n 1.10 A simplified proof of 1.10 was given by Goldberg in his Ph.D. thesis 47.

5 4 International Journal of Mathematics and Mathematical Sciences 1.3. Partitions Where No Odd Part Is Repeated Let podn denote the number of partitions of n where no odd part appears more than once. Let gx denote the generating function of podn, so we have gx podnx n n0 1 x 2m 1 fxf ) x 4. 1 x2m fx that Via another identity of Gauss see equation , page 23 in 44, 45, it turns out 1 gx x nn1/2 n0 1 n x 2n2 n ; n 1.12 so in this case the reciprocal of the generating function under consideration has nonzero coefficients at the onents which are triangular or equivalently, hexagonal numbers. The analogous Rademacher-type formula for podn is as follows: podn 2 ), h< h,1 ωh, ω4h/, 4,/, 4 e 2inh/ ω2h/, 2,/, 2 d sinh, 48n 1/4 dn 8n 1 ) Equation 1.13 is the case r 2ofTheorem 2.1 to be presented in the next section. 2. A Common Generalization Let us define f r x : fxf ) x 2r, 2.1 fx 2

6 International Journal of Mathematics and Mathematical Sciences 5 where r is a nonnegative integer. Thus, f 0 x fx pnx n, n0 f 1 x fx pnx n, n0 f 2 x gx podnx n. n0 2.2 Let p r n denote the coefficient of x n in the ansion of f r n, thatis, f r x p r nx n. n0 2.3 Notice that f r x can be represented by several forms of equivalent infinite products, each of which has a natural combinatorial interpretation: f r x 1 x m 1 x 2r m 1 ) 1 x 2m 1 1 x 2r m r 1 1 ) 1 x 2r 1 mλ x 2r 1 m λ1 Thus, p r n equals each of the following: i the number of overpartitions of n where nonoverlined parts are multiples of 2 r by 2.4; ii the number of partitions of n where all parts are either odd or multiples of 2 r by 2.5, provided r 1; iii the number of partitions of n where nonmultiples of 2 r 1 are distinct by 2.6, provided r 1.

7 6 International Journal of Mathematics and Mathematical Sciences Theorem 2.1. For r 0, 1, 2, 3, 4, p r n 2r1/2 3 0 h< h,1 1,2 maxr,1 1 e 2inh/ ωh, ω2r h, ω2h, d sinh 24n 2 r r 1) / 2 r/2 6 )) dn 24n 2r h< h,1 r j1 r/2 2 2 jr/2 1,2 r 2 j e 2inh/ ωh, ω 2 r j h, 2 j ) ωh, /2 d sinh 24n 2 r j r) ) /6 dn 24n 2r A Proof of Theorem 2.1 The method of proof is based on Rademacher s proof of 1.6 in 48 with the necessary modifications. Additional details of Rademacher s proof of 1.6 are provided in 49, 50, Chapter 14, and51, Chapter 5. Of fundamental importance is the path of integration to be used. In 48, Rademacher improved upon his original proof of 1.6 given in 3, by altering his path of integration from a carefully chosen circle to a more complicated path based on Ford circles, which in turn led to considerable simplifications later in the proof Farey Fractions The sequence F N of proper Farey fractions of order N is the set of all h/ with h, 1 and 0 h/ < 1, arranged in increasing order. Thus, for example, F 4 {0/1, 1/4, 1/3, 1/2, 2/3, 3/4}. For a given N, leth p, h s, p,and s be such that h p / p is the immediate predecessor of h/ and h s / s is the immediate successor of h/ in F N. It will be convenient to view each F N cyclically, that is, to view 0/1 as the immediate successor of N 1/N.

8 International Journal of Mathematics and Mathematical Sciences Ford Circles and the Rademacher Path Let h and be integers with h, 1and0 h<.theford circle 52 Ch, is the circle in C of radius 1/2 2 centered at the point h 1 2 i point The upper arc γh, of the Ford circle Ch, is those points of Ch, from the initial p α I h, : h 1 2 p 2 ) i 2 p to the terminal point s α T h, : h 1 2 s 2 ) i, 2 s traversed in the clocwise direction. Note that we have α I 0, 1 α T N 1,N. 3.4 Every Ford circle is in the upper half plane. For h 1 / 1,h 2 / 2 F N, Ch 1, 1 and Ch 2, 2 are either tangent or do not intersect. The Rademacher path PN of order N is the path in the upper half of the τ-plane from i to i 1 consisting of γh, h/ F N 3.5 traversed left to right and clocwise. In particular, we consider the left half of the Ford circle C0, 1 and the corresponding upper arc γ0, 1 to be translated to the right by 1 unit. This is legal given then periodicity of the function which is to be integrated over PN Set Up the Integral Let n and r be fixed, with n>2 r 1/24. Since f r x p r nx n, n0 3.6

9 8 International Journal of Mathematics and Mathematical Sciences Cauchy s residue theorem implies that p r n 1 f r x dx, 2i C x 3.7 n1 where C is any simply closed contour enclosing the origin and inside the unit circle. We introduce the change of variable x e 2iτ 3.8 so that the unit dis x 1inthex-plane maps to the infinitely tall, unit-wide strip in the τ-plane where 0 Rτ 1andIτ 0. The contour C is then taen to be the preimage of PN under the map x e 2iτ. Better yet, let us replace x with e 2iτ in 3.7 to ress the integration in the τ-plane: p r n f r e 2iτ) e 2inτ dτ PN f r e 2iτ) e 2inτ dτ h/ F N γh, N f r e 2iτ) e 2inτ dτ. γh, 1 0 h< h, Another Change of Variable Next, we change variables again, taing τ iz h, 3.10 so that z i τ h ) Thus Ch, in the τ-plane maps to the clocwise-oriented circle K centered at 1/2 with radius 1/2. So we now have in the z-plane N p r n i h< h,1 zt h, e 2inh/ e 2nz/ f r e 2ih/ 2z/) dz, z I h, arc of K 3.12 where z I h, resp., z T h, is the image of α I h, see 3.2 resp. α T h, see 3.2 under the transformation 3.11.

10 International Journal of Mathematics and Mathematical Sciences 9 So the transformation 3.11 maps the upper arc γh, of Ch, in the τ-plane to the arc on K which initiates at z I h, 2 p 2 p i 2 p and terminates at z T h, 2 s 2 s i. 2 s Exploiting a Modular Transformation From the theory of modular forms, we have the transformation formula 2, page 93, Lemma 4.31: 2ih f 2z )) z 1 z ) ) )) zf ωh, 2i iz 1 H, where z is the principal branch, h, 1, and H is a solution to the congruence hh 1mod From 3.15, we deduce the analogous transformation for f r x. The transformation formula is a piecewise defined function with r 1 cases corresponding to j 0, 1, 2,...,r, where, 2 r 2 j : 2ih f r 2z )) ωh, ω 2 r j h, 2 j ) ω 2h/ ) )) 2 δ j0,/ 2 δj j r1 ) ) 2 2 δ j0 24z 1 2r z 12 z 2 r j 1 ) 2 δ j0 f 2/z 2H j i/ )) f 2 2j r1 /z 2 2j r1 H j i/ )) f ) 2/z 2 δ j0 Hj ) )), 2i/ 2 δ j0 3.17

11 10 International Journal of Mathematics and Mathematical Sciences where H j is divisible by 2 r j and is a solution to the congruence hh j 1mod, and 1 if j 0, δ j0 0 if j / is the Kronecer δ-function. Notice that in particular, for r/2 j r, 3.17 simplifies to 2ih f r 2z )) ωh, ω 2 r j h, 2 j ) ωh, /2 2 z2 r j f 2j r z 2H )) ji. [ ) ] ) 2 2j r 1 z r z Since the r 0 case was established by Zucerman, and the r 1 case by Rademacher, we will proceed with the assumption that r>1. Apply 3.17 to 3.12 to obtain p r n i r N 1 e 2inh/ j0 1 0 h<,2 r j h,1 zt h, z I h, arc of K ωh, ω 2 r j h, 2 j ) ω 2h/ ) )) 2 δ j0,/ 2 δj j r1 2 δ j0 2) 24z 24n 1 2r z 12 z2 r j 1 ) 2 δ j0 f 2/z 2H j i/ )) f 2 2j r1 /z 2 2j r1 H j i/ )) )) dz. f 2 δ j0 2 /z H j 2 δ j0 2 i/ 3.20

12 International Journal of Mathematics and Mathematical Sciences Normalization Next, introduce a normalization ζ z this is not strictly necessary, but it will allow us in the sequel to quote various useful results directly from the literature: p r n i r N 5/2 e 2inh/ j0 1 0 h<,2 r j h,1 ζt h, ζ I h, arc of K ωh, ω 2 r j h, 2 j ) ω 2h/ ) )) 2 δ j0,/ 2 δj j r1 ) ) 2 2 δ j0 ζ2 r j 1 ) 2 δ j0 24ζ 24n 1 2r ζ 12 2 f 2/ζ 2H j i/ )) f 2 2j r1 /ζ 2 2j r1 H j i/ )) f ) 2/ζ 2 δ j0 Hj ) )) dζ, 2i/ 2 δ j where Let us now rewrite 3.21 as p r n i r N 2 ζ I h, p i, 2 p 2 2 p 2 ζ T h, s i. 2 s 2 2 s 2 5/2 j0 1 0 h<,2 maxr,1 2 j h,1 2 e 2inh/ ωh, ω 2 r j h, 2 j ) ω 2h/ 2 δ j0 ),/ 2 δj0 )) I j,1 I j,2 ), where I j,1 : ζt h, ζ I h, arc 2 2 2j r1 2 δ j0 2) ζ2 r j 1 ) 2 δ j0 24ζ 24n 1 2r ζ 12 2 f 2/ζ 2Hj i/ )) f 2 2j r1 /ζ 2 2j r1 H j i/ )) 1 )) f 2 δ j0 2 /ζ H j 2 δ j0 2 i/ dζ,

13 12 International Journal of Mathematics and Mathematical Sciences ζt h, 2 2 2j r1 2 δ j0 2) I j,2 : 24n 1 2r ζ 24ζ 12 2 ζ I h, arc 3.7. Estimation ζ2 r j 1 ) 2 δ j0 dζ It will turn out that as N,onlyI j,2 for j 0and r/2 <j r ultimately mae a contribution. Note that all the integrations in the ζ-plane occur on arcs and chords of the circle K of radius 1/2 centered at the point 1/2. So, inside and on K, 0< Rζ 1andR1/ζ Estimation of I j,2 for 1 j r/2 The regularity of the integrand allows us to alter the path of integration from the arc connecting ζ I h, and ζ T h, to the directed segment. By 51, page 104, Theorem 5.9, the length of the path of integration does not exceed 2 2/N, and on the segment connecting ζ I h, to ζ T h,, ζ < 2/N. Thus, the absolute value of the integrand is 2 2j r 1 ) 12ζ 24n 1 ) 2r ζ ζ2 r j n 1 2 ζ 1/2 2 r j/2 r ) Rζ 2 2j r 1 ) R 1 ) ζ ζ 1/2 2 r/2 2n Thus, for 1 j r/2, Ij,2 2 ) 1/ r/2 e 2n C j N N 3/2 N 3/ for a constant C j recalling that n and r are fixed Estimation of I j,1 for 1 j r/2 We have the absolute value of the integrand: 2 2j r 1 ) ζ2 r j 24n 1 ) 2r ζ 12ζ f 2/ζ 2H j i/ )) f 2 2j r1 /ζ 2 2j r1 H j i/ )) 4 f 4H )) ji ζ

14 International Journal of Mathematics and Mathematical Sciences j r 1 ) ζ2 r j 24n 1 ) 2r ζ 12ζ f 2j r 2 ζ 2H )) ji 24n 1 2 ζ 1/2 2 r j/2 r ) Rζ 2 2j r 1 ) R 1 ) ζ 2m p 2j r m ζ 2H ) jim 24n 1 2 ζ 1/2 2 r j/2 r ) p 2j r m 12 ζ 1/2 2 r/2 e 2n p 0 me 2m 12 )) 24m 2 2j r 1 c j ζ 1/ for a constant c j.so,for1 j r/2, Ij,1 2 ) 1/2 2 2 c j <C j 3/2 N 3/ N N for a constant C j Estimation of I 0,1 Let p rx be defined by prnx n f ) x 2r fx f 2r x 1) n0 Again, the regularity of the integrand allows us to alter the path of integration from the arc connecting ζ I h, and ζ T h, to the directed segment.

15 14 International Journal of Mathematics and Mathematical Sciences With this in mind, we estimate the absolute value of the integrand: ) r 24ζ 24n 1 ) 2r ζ ζ2 r f 2/ζ 2H0 i/ ) f 2 1 r /ζ 2 1 r H 0 i/ )) 1 f /ζ H 0 i/ ) ) r 24ζ 2 p 1 r m m ζ 24n 1 ) 2r ζ ζ2 r ) 21 r H 0 im ) r R 1 ) 24n 1 2 r ) Rζ ζ 1/2 2 r 1/2 24 ζ 12 2 ) ) 2 p 1 r m 2 1 r H 0 im m ζ 3.30 e 2n ζ 1/2 2 r 1/2 p m ) r R 1 24 ζ 21 r mr 1 ) ζ e 2n ζ 1/2 2 r 1/2 p m 24m 24 2 R1 1 2 r 1)) r 1 ζ e 2n ζ 1/2 2 r 1/2 p m 24m 1 2 r 1)) 24 <e 2n ζ 1/2 2 r 1/2 p 24m 1 2 r 1) y 24m 1 2 r 1 where y e /24) c 0 ζ 1/2 for a constant c 0.So, I 0,1 2 2 N ) 1/2 2 c 0 <C 0 3/2 N 3/ N for a constant C 0.

16 International Journal of Mathematics and Mathematical Sciences Estimation of I j,1 for 1 r/2 j r 4 Again, the regularity of the integrand allows us to alter the path of integration from the arc connecting ζ I h, and ζ T h, to the directed segment. With this in mind, 2 2j r 1 ) 12ζ 1 f 2j r 24n 1 ) 2r ζ ζ2 r j ζ 2H )) ji 2 2j r 1 ) ) 24n 1 2 r ) ζ ζ2 r j 12ζ 12 2 ) ) 2m 2Hj im p 2j r m ζ 2 2j r 1 ) R 1 ) 24n 1 2 r ) Rζ ζ 1/2 2 r j/2 12 ζ p 2j r m 2mR 1 ) ) 2Hj im ζ e 2n ζ 1/2 2 r/2 p 2j r m 12 R1 ζ ) ) 24m 2 2j r 1 e 2n ζ 1/2 2 r/2 p 0 m ) 12 R1 ζ 24m 2r 1 c j ζ 1/2 for a constant c j.so, Ij,1 2 ) 1/2 2 2 c j <C j 3/2 N 3/ N N for a constant C j, when 1 r/2 j r.

17 16 International Journal of Mathematics and Mathematical Sciences Combining the Estimates One has i N 1,2 r 1 r/2 i i j1 5/2 N 0 h< h,1 1,2 r 2 j r N e 2inh/ ωh, ω2r h, I 0,1 ω2h, 5/2 0 h< h,1 5/2 j1 r/2 1 0 h<,2 r 2 j h,1 e 2inh/ ωh, ω 2 r j h, 2 j ) ) Ij,1 I j,2 ωh, /2 e 2inh/ ωh, ω 2 r j h, 2 j ) ωh, /2 I j, < r N 1 C j 1 N 3/2 j0 1 h0 C N 3 2 N 1, 1 O N 1/2). r/2 j1 where C N 1 C j 1 N 3/2 1 h0 r r/2 C j j0 C j j1 Thus, we may revise 3.21 to N p r n i 1 i 5/2 r 0 h< h,1 N j1 r/2 1 e 2inh/ ωh, ω2r h, I 0,2 ω2h, 5/2 0 h< h,1 e 2inh/ ωh, ω 2 r j h, 2 j ) I j,2 O N 1/2). ωh, / Evaluation of I j,2 for j 0 and 1 r/2 j r Write I j,2 as 2 2j r 1 ) I j,2 K 12ζ 24n 1 ) 2r ζ ζ2 r j dζ I 12 2 j,3 I j,4, 3.36

18 International Journal of Mathematics and Mathematical Sciences 17 where I j,3 : ζi h,, I j,4 : 0 0 ζ T h,, 3.37 and I j,3 and I j,4 have the same integrand as 3.36and analogously for I 0, Estimation of I j,3 and I j,4 We note that the length of the arc of integration in I j,3 is less than / 2N, andonthis arc ζ < 2/N. 50, page 272.Also,R1/ζ 1onK 50, page 271, equation Further, 0 < Rζ <2 2 /N 2 50, page 271, equation The absolute value of the integrand is thus 2 r j ζ 1/2 24n 1 2 r Rζ 2 2j r 1 ) R 1 ) ζ 24n 1 2 < 2 r j/2 2 1/4 1/2 N 1/2 r 2 2j r 1 ) ) N 2 12 so that Ij,2 24n 1 2 <2 1/2 N 1 2 r j/2 2 1/4 1/2 N 1/2 r 2 2j r 1 ) ) 6N n 1 2 3/2 N 3/2 2 2r 2j 1/4 r 2 2j r 1 ) ) 6N n 1 2 O 3/2 N 3/2 r )). 6N By the same reasoning, I j,3 O 3/2 N 3/2 24n 1 2 r /6N 2. We may therefore revise 3.35 to N p r n i 1 5/2 0 h< h,1 e 2inh/ ωh, ω2r h, ω2h, { 2 1 r 1 ) ζ2 r 1 K 24ζ i r N j1 r/2 1 5/2 0 h< h,1 { 2 2j r 1 ) ζ2 r j K 12ζ 24n 1 } 2r ζ dζ 12 2 e 2inh/ ωh, ω 2 r j h, 2 j ) ωh, /2 24n 1 } 2r ζ dζ O N 1/2)

19 18 International Journal of Mathematics and Mathematical Sciences and upon letting N tend to infinity, obtain p r n i 3.9. The Final Form 1 5/2 0 h< h,1 e 2inh/ ωh, ω2r h, ω2h, { 2 1 r 1 ) ζ2 r 1 K 24ζ i r j1 r/2 1 5/2 0 h< h,1 { 2 2j r 1 ) ζ2 r j K 12ζ We may now introduce the change of variable ζ 24n 1 } 2r ζ dζ 12 2 e 2inh/ ωh, ω 2 r j h, 2 j ) ωh, / j r1 ) ) 2 2 δ j0 24t 24n 1 } 2r ζ dζ where the first summation in 3.41 is the j 0 term separated out for clarity, which allows the integral to be evaluated in terms of I 3/2, the Bessel function of the first ind of order 3/2 with purely imaginary argument 53, page 372, 17.7 when we bear in mind that a bent path of integration is allowable according to the remar preceding Equation 8 on page 177 of 54. Seealso51, page 109. The final form of the formula is then obtained by using the fact that Bessel functions of half-odd integer order can be ressed in terms of elementary functions. We therefore have, for c 2 2 2j r1 δ j0 2 /24, p r n i 1 i 5/2 ci c i 0 h< h,1 e 2inh/ ωh, ) 3/2 ω2r h, 2 r 1/2 21 r 1 ω2h, 24 { ) } r t 5/2 24n 1 2 r 2 t dt t r j 1 r/2 1 0 h< h,1 ci c i 5/2 e 2inh/ ωh, ω 2 r j h, 2 j ) ωh, /2 ) 3/2 2 r j/2 22j r 1 12 { 2 2j r t 5/2 1 ) } 24n 2 r 1 2 t dt t

20 International Journal of Mathematics and Mathematical Sciences 19 24n 2 r 1 3/4 ) 1 2 r 1 3/4 1 2 r 2/4 1 0 h< h,1 r j1 r/2 0 h< h,1, 2 maxr,1 1 e 2inh/ ωh, ω2r h, ω2h, ) 3/42 2 2j r 2 jr/2 1 I 3/2 1,2 r 2 j e 2inh/ ωh, ω 2 r j h, 2 j ) ωh, /2 24n 2 r r 1) 2 r/2 6 1 I 3/2 24n 2 r j r), which, after application of the formula 51, page 110 I 3/2 z 2z d sinh z dz z ), 3.44 is equivalent to Theorem 2.1. Acnowledgments The author thans the anonymous referee for bringing the wor of Zucerman 46 and Goldberg 47 to his attention. This, in turn, led the author to see the more general result presented here in this final version of the paper. References 1 L. Euler, Introductio in Analysin Infinatorum, Marcum-Michaelem Bousquet, Lausanne, Switzerland, G. H. Hardy and S. Ramanujan, Asymptotic formulae in combinatory analysis, Proceedings of the London Mathematical Society, vol. 17, pp , H. Rademacher, On the partition function pn, Proceedings of the London Mathematical Society, vol. 43, pp , E. Grosswald, Some theorems concerning partitions, Transactions of the American Mathematical Society, vol. 89, pp , E. Grosswald, Partitions into prime powers, The Michigan Mathematical Journal, vol. 7, pp , 1960.

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