A Hardy-Ramanujan-Rademacher-type formula for (r, s)-regular partitions

Transcription

1 The Ramanujan Journal manuscript No. will be inserted by the editor A Hardy-Ramanujan-Rademacher-type formula for r, s-regular partitions James Mc Laughlin Scott Parsell Received: date / Accepted: date Abstract Let p r,s n denote the number of partitions of a positive integer n into parts containing no multiples of r or s, where r > 1 and s > 1 are square-free, relatively prime integers. We use classical methods to derive a Hardy-Ramanujan-Rademacher-type infinite series for p r,s n. Keywords q-series partitions circle-method Hardy-Ramanujan- Rademacher Mathematics Subject Classification 2000 Primary 11P82 Secondary 05A17 11L05 11D85 11P55 11Y35 1 Introduction A partition of a positive integer n is a representation of n as a sum of positive integers, where the order of the summands does not matter. We use pn to denote the number of partitions of n, so that, for example, p4 = 5, since 4 may be represented as 4, 3 + 1, 2 + 2, and The function pn increases rapidly with n, and it is difficult to compute pn directly for large n. Rademacher [23], by slightly modifying earlier wor of Hardy and Ramanujan [13], derived a remarable infinite series for pn. To describe this series This wor was partially supported by a grant from the Simons oundation # to James Mc Laughlin. The second author is supported in part by National Security Agency grant H J. Mc Laughlin S. Parsell Mathematics Department, West Chester University, 25 University Avenue, West Chester, PA jmclaughl@wcupa.edu, sparsell@wcupa.edu

2 2 James Mc Laughlin, Scott Parsell we need some notation. Recall that the Dedeind sum se, f is defined by se, f := f 1 r=1 r er f f er 1, f 2 and for ease of notation, we use ωe, f to denote expπi se, f, and for a positive integer, set A n := ωh, e 2πinh/. 0 h< h,=1 Recall also that I ν z denotes the modified Bessel function of the first ind. Theorem 1 Rademacher If n is a positive integer, then 2π A n π 2 pn = I 24n 1 3/4 3/2 n =1 Rademacher s series converges incredibly fast. or example, p500 = 2, 300, 165, 032, 574, 323, 995, 027, and yet six terms of the series are sufficient to get within 0.5 of p500. The idea of course is that if a partial sum is nown to be within 0.5 of the value of the series, then the nearest integer gives the exact value of pn. Since the publication of Rademacher s paper [23], a number of authors have found series similar to 1.1 for certain restricted partition functions. Lehner [19] found such series for p 1 n and p 2 n, the number of partitions of n into parts ±1 mod 5 and ±2 mod 5 respectively, and this was extended by Livingood [20] to series for p 1 n,..., p q 1/2 n, the number of partitions into parts ±1 mod q, ±2 mod q,..., ±q 1/2 mod q respectively, where q > 3 is an odd prime. Hua [14] derived a Rademacher-type series for p O n, the number of partitions of n into odd parts. Let q 3 be an odd prime and a = {a 1, a 2,..., a r } be a set of distinct integers satisfying 1 a i q 1/2. Hagis [5] gave a Hardy-Ramanujan- Rademacher-type series H.R.R. series for p a n, the number of partitions of n into parts ±a i mod q. In a subsequent series of papers [6 12], Hagis also developed similar series for other restricted partition functions into odd parts, odd distinct parts, no part repeated more than t times, etc.. Niven [21] gave a H.R.R. series for p 2,3 n, the number of partitions of n into parts containing no multiples of 2 or 3. In a similar vein, Haberzetle [4] gave a series for p q1,q 2 n, the number of partitions of n into parts containing no multiples of q 1 or q 2, where q 1 and q 2 are distinct primes such that 24 q 1 1q 2 1. Isei [15 17] derived H.R.R. series that, amongst other results, extended the result of Livingood [20] cited above from a prime q to a composite integer M, and also extended the results of Niven [21] and Haberzetle [4], by finding

3 A Hardy-Ramanujan-Rademacher-type formula for r, s-regular partitions 3 a H.R.R. series for p M n, the number of partitions of n into parts relatively prime to a square-free positive integer M. Sastri et al. [22,24,25] derived a number of H.R.R. series which, amongst other results, extended the result of Hagis cited above from a prime q to an arbitrary positive integer m. More recently, Sills [26 28] has partly automated the process of finding H.R.R. series for restricted partition functions, and aided by the use of the computer algebra system Mathematica, has found many new such series, including ones for restricted partition functions represented by various identities of Rogers-Ramanujan type. When r > 1 and s > 1 are relatively prime integers, let p r,s n denote the number of partitions of n into parts containing no multiples of r or s. We say that such a partition of an integer n is r, s-regular. In the present paper we give a H.R.R. series for p r,s n when r and s are square-free. We note that this result includes those Niven [21] and Haberzetle [4] as special cases. We now state our result explicitly. Define τ = 1 =1 1 e2πiτ, and denote by H i,j a solution to the congruence ih i,j 1 mod j, and for consistency of notation below, set H 0,1 = 0. or integers, r and s, let r := gcdr, and s := gcds, and, for ease of notation, set R := r 1s 1 24 Our result may be stated as follows., δ := r/r r s/s s Theorem 2 Let r > 1 and s > 1 be square-free relatively prime integers. or a positive integer and non-negative integer h with h, = 1, define the sequence {c m h, } by Hh, + i z Hhr/r,/r /r If n > R, then p r,s n = where A,m n := Hhrs/r s,/r s /r s + ir2 s2 rsz + ir2 Hhs/s,/s rz /s + is2 sz δ =1 m=0 1 h=0 h,=1 2πA,m n := 2πmr s c m h, exp. rsz m=0 r s δ m 4π r s I 1 rsn R rs δ m n R, ωh, ωhrs/r s, /r s ωhr/r, /r ωhs/s, /s c mh, exp 2πinh.

4 4 James Mc Laughlin, Scott Parsell The method of proof follows to a large extent the method used by previous authors to derive similar convergent series for other partition functions. In section 2, the Cauchy Residue Theorem is applied to the generating function for the sequence p r,s n, and a change of variable is then applied to convert the path of integration to the line segment [i, i + 1]. Next, this line segment is deformed to follow the path along the top of a collection of ord circles, after which another change of variable transforms the arc along the top of each ord circle to an arc along the circle in the complex plain with center 1/2 and radius 1/2. Next, the transformation formula for the Dedeind eta function ητ is used to transform the integrand into a form whose properties can be exploited to derive the final series stated in Theorem 2. Each transformed infinite product is expanded in a series, which is broen into an initial finite part which eventually leads to the series of the theorem and a tail, whose contribution is shown to be negligible. The path of integration for each of the terms coming from the tail of the series mentioned above is divided into three arcs. In section 3, Kloosterman sum estimates are developed, which are used in section 4 to get error bounds on the integrals along the three arcs for each term in the tail. This shows that these error terms go to zero as N, where N is the order of the arey sequence giving rise to the collection of ord circles. In section 5, the arcs of integration along the circle with center 1/2 and radius 1/2 for the main terms are replaced with a new path along the entire circle. It is shown that the contributions from the additional arcs also go to zero as N, where N is as in the paragraph above. Two other changes of variable and an application of an integral formula for modified Bessel functions of the first ind lead the final result. Remar: With the notation for τ as above and for ητ as below, the generating functions πir 1s 1τ/12 τ rsτ e rτ sτ = ηrτηsτ ητηrsτ are weight-zero modular forms, so that the general theorem of Bringmann and Ono [3] could in theory be used to derive our series for p r,s n. However, we prefer to employ the Hardy-Ramanujan-Rademacher method. 2 Initial transformations Let Gx = n=0 p r,s nx n = xr ; x r x s ; x s x; x x rs ; x rs 2.1 denote the generating function for the sequence {p r,s n}. By the Cauchy Residue Theorem, p r,s n = 1 Gx dx, 2πi C xn+1

5 A Hardy-Ramanujan-Rademacher-type formula for r, s-regular partitions 5 where C is any positively oriented simple closed curve inside the unit circle containing the origin. As usual, we start by taing C to be the circle centered at the origin with radius e 2π, and mae the change of variable x = e 2πiτ to get p r,s n = i+1 i Ge 2πiτ e 2πinτ dτ. We follow Rademacher by deforming the path of integration so that it traces the upper arcs of the collection of ord circles { C h, : h } N, where C h, is the circle with center h/ + i/2 2 and radius 1/2 2, and N is the set of arey fractions of order N. We denote the part of the path that is an arc of the circle C h, by γh,. Thus p r,s n = N 1 =1 h=0 h,=1 γh, τ rsτe 2πinτ rτ sτ dτ. 2.2 Next, for each circle C h,, set z = i 2 τ h/, transforming the circle C h, to the circle K with center 1/2 and radius 1/2, and transforming the arc γh, to the arc not passing through 0 on the latter circle joining the points z 1 h, = 2 + i and z 2 h, = 2 i , where h 1 / 1 < h/ < h 2 / 2 are consecutive arey fractions in N. With these changes, p r,s n = N 1 =1 h=0 h,=1 z2h, z 1 h, h + iz rs h 2 + iz e 2πin h 2 + iz 2 i r h + iz s h 2 + iz dz. 2 2 Next, recall that the Dedeind eta function is defined by ητ = e πiτ/12 =1 1 e 2πiτ 2.3 and satisfies the transformation formula see for example Apostol [1], Theorem 3.4 η aτ + b a + d = exp πi + s d, c { icτ + d} 1/2 ητ 2.4 cτ + d 12c

6 6 James Mc Laughlin, Scott Parsell a b whenever is an element of the modular group, c > 0, and τ lies in the c d upper half-plane. Thus τ = πi 12 exp τ aτ + b cτ + d + a + d c exp πis d, c { icτ+d} 1/2 aτ + b. cτ + d In what follows, for each set of choices for a, c and d, we tae b to be ad 1/c. or v {1, r, s, rs},, rs = r s and τ = h/ + iz/ 2, we set v = v,, so that v, v {1, 1, r, r, s, s, rs, r s }. We then transform vτ by setting c = /v, d = hv/v and a = H hv/v,/v, to get vh + ivz πv 2 2 = exp 12vz πvz hv 12 2 exp πi s, v v { } 1/2 vz Hhv/v,/v + iv v /v vz On substituting into 2.3, this gives p r,s n = N 1 =1 h=0 h,=1 z2 h, ωh, ωhrs/r s, /r s ωhr/r, /r ωhs/s, /s exp exp z 1 h, Hh, + i z Hhr/r,/r /r r s δ 2π rsz n Rz + 2πinh 2 Hhrs/r s,/r s /r s + ir2 s2 rsz + ir2 Hhs/s,/s rz /s + is2 sz i 2 dz. 2.6 We temporarily fix v {1, r, s, rs} and introduce the shorthand g = v = v,. We observe that the congruences H vh/g,/g vh/g 1 mod /g and H h, h 1 mod imply that vh vh/g,/g gh h, mod when h, = 1. Since r and s are square-free, we have v/g, = 1, and hence the congruence hv/g H h, 1 mod has a solution H h,, and we are free to tae H h, = v/g H h, to be a multiple of v/g. In particular, then, one has vh vh/g,/g gh h, mod v, and since v/g, = 1 it follows from 2 and the Chinese Remainder Theorem that vh vh/g,/g gh h, mod v/g.

7 A Hardy-Ramanujan-Rademacher-type formula for r, s-regular partitions 7 Hence the periodicity of τ implies that ghvh/g,/g + ig2 = vz g 2 H h, v + ig2. vz Put µ z = r s rs Hh, + i z. Then we deduce from 2 that the ratio appearing in 2.6 is where we write rs/r s µ z r s µ z r s/s µ z s r/r µ z := G µ z, 2.7 G τ = c m, exp2πimτ 2.8 m=0 for some coefficients c m,. We note that the coefficients c m h, occurring in the statement of Theorem 2 satisfy 2πimr s H h, c m h, = c m, exp, rs so that in particular c m h, = c m,. Then 2.6 may be expressed as p r,s n = where h, h,=1 i 2 Ω h,e 2πinh/ z2h, z 1 h, r s δ exp 2π + rsz n Rz 2 G µ z dz, Ω h, = ωh, ωrsh/r s, /r s ωrh/r, /r ωsh/s, /s. 2.9 We further introduce the notation 2πr s δ m Ψ m, z = exp + rsz which allows us to write p r,s n = N =1 i 2 m=0 c m, 0 h 1 h,=1 2πn Rz 2, πi Ω h, exp rs r s mh h, rsnh z2 h, z 1h, Ψ m, z dz. 2.11

8 8 James Mc Laughlin, Scott Parsell We decompose the sum over m into two parts, m < δ and m δ, and write p r,s n = P 1 n; N + P 2 n; N 2.12 for the resulting decomposition of We aim to show that P 2 n; N contributes a negligible amount to the formula. We find it useful to split the path of integration from z 1 h, to z 2 h, into the three arcs [z 1, zn], [z N, z 2 ], and [z N, zn], and we further decompose the first two as unions of arcs of the shape [zl, zl + 1], where zl = l 2 + il 2 + l It is easy to chec that each zl lies on the circle z 1/2 = 1/2. Since 1,, 2 are denominators of consecutive elements in the arey sequence of order N, we have + 1 N + 1 and + 2 N + 1, and hence 1 N + 1 and 2 N 1 for all values of h and. Moreover, since h 1 h 1 = h 2 h 2 = 1, we have h 1 1 mod and h 2 1 mod. It follows that H h, 2 1 mod, and hence the condition 2 l 1 1 is equivalent to a restriction of H h, to some interval I l modulo. We may therefore interchange the order of summation and integration in 2.11 to obtain P 2 n; N = N =1 i 2 m= δ c m, S 1 m, + S 2 m, + S 3 m,, 2.14 where S 1 = N 1 l=n+1 zl+1 zl Ψ m, zθ, l, m dz, S 2 = N 2 l= N S 3 = zl+1 zl zn z N Ψ m, zθ, l, m dz, Ψ m, zθ, l, m dz, and Θ, l, m = 0 h 1 h,=1 H h, I l 2πi Ω h, exp rs r s mh h, rsnh. In order to mae further progress, we must develop suitable estimates for Θ, l, m. We tae up this tas in the next section.

9 A Hardy-Ramanujan-Rademacher-type formula for r, s-regular partitions 9 3 Kloosterman sums In order to estimate S 1, S 2, and S 3, we aim to express Θ, l, m in terms of Kloosterman sums. As a first step, we are able to remove the restriction on H h, in the summation at a cost of Olog. The argument is similar to that of Hagis [11] see also Lehner [19]. Lemma 1 or each and m, there exists an integer j = j, m with 0 j 1 such that for every l one has 1 Θ, l, m 1+log 2πi Ω h, exp rs r s m + rsjh h, rsnh, h=0 h,=1 where the implicit constant is absolute. Proof ix, l, and m, and let I l = [α, β], where α and β are integers with 0 β α <. By orthogonality, we have 1 1 { 2πijH t 1 if H t mod exp = 0 if H t mod, j=0 and hence the expression j=0 1 β 1 2πijH t exp t=α j=0 is 1 if H is congruent mod to one of the integers in I l and 0 otherwise. We therefore have Θ, l, m = 1 1 2πi γ j Ω h, exp rs r s m + rsjh h, rsnh, where 0 h 1 h,=1 γ j = β 2πijt exp. t=α By summing this geometric progression, we find that 1 γ j min β α, min, 1 sinπj/ 2 j/ 1, 3.1 where denotes the distance to the nearest integer. One now easily gets see for example Lemma 3.2 of Baer [2] 1 γ j 1 + log, j=0

10 10 James Mc Laughlin, Scott Parsell and the lemma follows after taing the maximum over j in the inner summation of 3.1. Write 24rs = AB, where A is the largest divisor of 24rs relatively prime to, and let Ā denote the multiplicative inverse of A modulo B. We note that every prime factor of B is a prime factor of, whence gcdh, = 1 if and only if gcdh, B = 1. Moreover, for each such h and we can find H h,b with the property that hh h,b 1 mod B and A H h,b. These observations allow us to calculate the Ω h, defined by 2.9 rather explicitly. Lemma 2 Suppose that h, = 1, let A, B, and H h,b be as above, and additionally write ν = r 1s 1, σ = r r s s, and 48πi Ā Φ h, = exp B 2 R r s δ H h,b, where R and δ are as in 1.2. When is odd one has r/r s/s Ω h, = exp s r πiν 4r s 2πi exp 2 δ r s R h Φ h,, r s and when is even one has s r 2πi Ω h, = exp 2 2 δ + 1 r/r s/s r s 8 σ + r s R h Φ h,. Proof When v {1, r, s, rs}, write g = v, and 2πi 2 g 2 ω v h, = exp 2hvg + v 2 24g 3 h 2 g 2 H vh/g,/g. 3.2 Then when is odd, formula 2.4 of Niven [21] gives hv/g πi ωvh/g, /g = exp 1 ω v h,. 3.3 /g 4 When is even, the condition that v is square-free implies that hv/g is odd, and hence we may apply formula 2.3 of [21] to obtain /g πi ωvh/g, /g = exp 2 hv + g ω hv/g 4 g2 v h,. 3.4 Since v is square-free, we have vh/g, B = 1. Therefore, as in the argument preceding the statement of the lemma, we may replace each H vh/g,/g in 3.2 by an integer H hv/g,b/g divisible by A, and the argument leading to 2 then gives H hv/g,b/g g v H h,b mod B/g,

11 A Hardy-Ramanujan-Rademacher-type formula for r, s-regular partitions 11 where we recall that H h,b is divisible by A and hence by v/g. Substituting into 3.2 now gives πi 2πi ω v h, = exp 6g 2 2 g 2 Ārs vh exp Bvg 2 2 g 2 vh 2 g 2 H h,b πi 2πi = exp 12g 2 2 g 2 Ārs vh exp Bv 2 g 2 H h,b, upon noting that v rs, g 2 2 g 2, and h 2 H h,b h mod B. It now follows with a bit of computation that ω 1 h, ω rs h, 2πi ω r h, ω s h, = exp R 2 δ hv Φ h,. r s The lemma now follows from 3.3 and 3.4 via routine calculations using the multiplicative properties of the Jacobi symbol. We now show that the summation on the right hand side of Lemma 1 is a Kloosterman sum with modulus B. ix j = j, m to be the integer in the statement of Lemma 1 for which the expression on the right is maximal and write T, m for the corresponding sum, so that for each l one has 0 h B 1 h,b=1 Θ, l, m 1 + log T, m. 3.5 rom the definition of the Dedeind sum see Section 1, together with 2.9, we see that Ω h+t, = Ω h, for all t Z. Hence we can write 2πi T, m = B 1 Ω h, exp 24 Ār s m + jbh h,b Bnh, B since the definition of B implies that h, = 1 if and only if h, B = 1. Moreover, since Ah runs over a reduced residue system modulo B as h does and since h 1 AH Ah,B H h,b mod B, we find that T, m = B 1 0 h B 1 h,b=1 Ω Ah, exp 2πi 24 Ār s m + B jbāh h,b ABnh. We are now able to express T, m in terms of the Kloosterman sum Ka, b; c = 2πiax + b x exp, c 1 x c x,c=1 3.6 where xx 1 mod c. In our case c = B, and H h,b plays the role of x. According to Weil s bound see for example Iwaniec and Kowalsi [18], Corollary one has Ka, b; c a, b, c 1/2 c 1/2+ε, 3.7 and this delivers the bound on Θ, l, m recorded in the following lemma.

12 12 James Mc Laughlin, Scott Parsell Lemma 3 One has Θ, l, m 1/2+ε, where the implicit constant depends at most on ε, r, s, and n. Proof On substituting the results of Lemma 2 with h replaced by Ah into 3.6, we obtain T m, = B 1 Ka, b; B where and b = 24Ā2 r s δ m 2 R jāb, α a = rs + 24R ABn, r s where α = 24 2 δ if is odd and α = 48 2 δ + 3σ if is even. Since r s and α, any common divisor of a and B must also divide the integer u = 24 ABn 24rsR = 576rsn R. In view of the hypothesis that n > R, we have u 0 and hence a, b, B r,s,n 1. The lemma now follows from 3.5 and The error terms In order to complete the analysis of P 2 n; N, we require an estimate for the growth rate of the coefficients c m, in 2.11 arising from the expansion 2.8. The following crude bound will suffice for our purposes. Lemma 4 One has c m, e 2π m, where the implicit constant is independent of. Proof or simplicity, we consider the series gx = c m, x m, m=0 so that 2.8 gives G τ = ge 2πiτ. Then by 2.7 one has gx = xa ; x a x b ; x b x c ; x c x d ; x d, where a = r s/s, b = s r/r, c = rs/r s, and d = r s. We have 1 x t ; x t = plx tl, l=0

13 A Hardy-Ramanujan-Rademacher-type formula for r, s-regular partitions 13 and it follows that the coefficient of x m in [x c ; x c x d ; x d ] 1 is bounded above by m + 1pm 2. urthermore, by Euler s Pentagonal number theorem we have x t ; x t = 1 l x tl3l 1/2, l= and from this one sees that the coefficient of x m in x a ; x a x b ; x b has absolute value at most 4 m + 2. Hence on applying the well-nown Hardy- Ramanujan asymptotic formula [13] for pm, we deduce that and the lemma follows. c m, m 5/2 pm 2 m 1/2 e 2π2/31/2 m e 2π m, We are now able to show that the terms in 2.11 with m δ contribute a negligible amount. irst of all, it follows from Lemma 3 and the definitions at the end of Section 2 that S 1 1/2+ε zn and zn+1 z N 1 Ψ m, z dz, S 2 1/2+ε Ψ m, z dz, zn S 3 1/2+ε Ψ m, z dz. z N z N Since Rez 1 in z 1/2 1/2, the definition 2.10 immediately gives 2πr s Ψ m, z n exp δ m. 4.1 rsz Moreover, one has Re1/z 1 in the dis z 1/2 1/2 and it follows that Ψ m, z 1 for all m δ. If δ < 0 then 4.1 yields Ψ m, z e 2πm/rs, whereas if δ > 0 and m > 2δ then we obtain Ψ m, z e πm/rs. With the above estimates in hand, it remains to bound the lengths of the various arcs of integration. After recalling 2.13, a simple calculation reveals that zn zn /N 2 and z N 1 z N 2 /N 2, while zn z N 1/N. Therefore, on shifting the paths of integration from the circle to the respective chords connecting the endpoints, we deduce from 2.14, Lemma 4, and the discussion following 4.1 that N P 2 n; N N 2 1/2+ε + N 1 3/2+ε 1 + =1 m>2δ c m, e πm/rs N 1/2+ε. 4.2 Thus on recalling 2.12 we get and hence it suffices to analyze P 1 n; N. p r,s n = P 1 n; N + ON 1/2+ε, 4.3

14 14 James Mc Laughlin, Scott Parsell 5 The main terms or each, we now consider the main terms if any with 0 m < δ. Recall that K is the circle with center 1/2 and radius 1/2, and let K denote this circle traversed in the clocwise direction. We write z2 h, z1 h, 0 =, z 1h, K 0 z 2h, and use this to decompose each of the integrals in Our aim is to show that the integrals over the arcs [z 1 h,, z 2 h, ] in 2.11 can be replaced by integration over K, with negligible error. By repeating the argument leading to 2.14, we find that the contribution from z 1 h, and 0 0 z 2h, is at most P 3 n; N = N = m<δ S 1, m + S 2, m + S 3, m. 5.1 Since the coefficient of 1/z in the exponent of Ψ m, z is positive when m < δ, we eep the path of integration on the circle, where we have Re1/z = 1, and hence 4.1 gives Ψ m, z 1. inally, it is easy to show see for example the proof of Apostol [1], Theorem 5.9 that each of the arcs [zn + 1, zn], [z N, z N 1], and [z N, zn] has length O/N. It therefore follows from 5.1 and Lemma 3 that P 3 n; N N 1 N =1 On letting N, we deduce from 4.3 and 5.2 that δ p r,s n = A,m n i =1 m=0 2 K 1/2+ε N 1/2+ε πr s exp δ m + 2πz n R dz, rsz 2 where A,m n is as in the statement of Theorem 2. It remains to express the integral over K in terms of modified Bessel functions of the first ind. Setting w = 1/z gives p r,s n = We now set =1 2 0 m<δ A,m n i 1+i 1 i t = 2πwr s rs 2πwr s exp δ m + 2π 1 rs 2 n R w w 2 dw. δ m and c = 2πr s rs δ m

15 A Hardy-Ramanujan-Rademacher-type formula for r, s-regular partitions 15 to get p r,s n = A,m n 2πc 2 0 m<δ =1 1 2πi c+i c i Lastly, we use the formula t 2 exp I ν z = z/2ν 2πi see Watson [29] with ν = 1 and to get, after some simplification, p r,s n = =1 t + 4π2 r s 2 δ m n R 1 dt. 5.3 rs t c+i c i t ν 1 exp t + z2 dt 4t [ z 4π 2 ] 1/2 2 = r s 2 δ m n R rs 2πA,mn 0 m<δ r s δ m rsn R I 1 4π The proof of Theorem 2 is now complete. r s rs δ m n R Convergence behaviour Obtaining a bound for the error in using the Nth partial sum in Rademacher s series to estimate pn is a difficult problem, and bounding the error in using the Nth partial sum of the series at 5.4 to estimate p r,s n is liely to be at least as difficult. We do not attempt an analysis of this problem in the present paper. However, we do examine a particular numerical example, to get a feel for the speed and the nature of the convergence. In the case examined r = 14, s = 15, n = 500, the convergence of the series is initially very fast, while it seems that once the partial sums of the series get to within 1.0 of the correct value, that convergence then proceeds much more slowly, with for rs the greatest contributions to the sum of the series coming from those terms with 0 mod rs, and with the contributions from the terms for the other being negligible in comparison. However, it is possible that the convergence behaviour may be different, if r and s have a different number of prime factors than in the example. As an illustration of the convergence behaviour, we consider the convergence of the sum of the series to p 14, = 310, 093, 947, 025, 073, 675, 623,

16 16 James Mc Laughlin, Scott Parsell Table 1 N S N p 14, S N by examining the difference p 14, S N, where S N is the Nth partial sum of the series. We tabulate the values for 1 N 11 in Table 6 to show the very fast convergence initially. Note that the terms in the series corresponding to = 5, = 7 and = 10 are zero, since δ 5, δ 7, δ 10 < 0, so that each of the inner sums over m are empty, and thus contribute zero to the value of the series the term in the series corresponding to = 8 is also zero, but this is because the terms in the inner sum over m add to zero. We next plot igure 1 the values for 1 N 750 the large initial values lie outside the range of the plot to show how the terms in the sum corresponding to = 210, 420, 630,... multiples of r s = contribute much more to the value of tail of the series than values of 0 mod rs. p 14, S N N ig. 1 The convergence of the series to p 14, The plot shows the values of p 14, S N, where S N is the Nth partial sum of the series. Note the jumps in the values of the partial sums for = 210, 420 and 630.

17 A Hardy-Ramanujan-Rademacher-type formula for r, s-regular partitions 17 p 500 SN N ig. 2 The convergence of Rademacher s series to p500. Note the more erratic convergence behaviour, compared that of the series for p 14, We remar that this apparent step-lie convergence behaviour of the series for p r,s n is in contrast to the apparent convergence behaviour of Rademacher s series for pn, which is more erratic. igure 2 is a plot of the difference p500 S N, 1 N 750, where S N is the Nth partial sum of Rademacher s series. p6, SN N ig. 3 The series corresponding to p 6, appears to converge to p 6,25 500, despite the fact that 25 is a square. We conclude by remaring that experimental evidence suggests that the requirement that r and s be square-free may be dropped, although it is not possible to employ the arguments used to get the Kloosterman sum estimates

18 18 James Mc Laughlin, Scott Parsell in this case. or example, seven terms of the series for p 6, = 42, 305, 606, 435, 448, 427, 065 appear to be sufficient to get within 0.5 of p 6, igure 3 is a plot of difference p 6, S N, 1 N 1600, where once again S N is the Nth partial sum of the series in Theorem 2. Note that the convergence of the series for p 6, exhibits the same step-lie behaviour seen above in the convergence of the series for p 14,15 500, with the steps this time being multiples of 150 = References 1. T. M. Apostol, Modular functions and Dirichlet series in number theory, Springer, R. C. Baer, Diophantine inequalities, Clarendon Press, Oxford, K. Bringmann and K. Ono, Coefficients of harmonic Maass forms, Proceedings of the 2008 University of lorida Conference on Partitions, q-series, and Modular orms, Developments in Mathematics series, Springer, to appear. 4. M. Haberzetle, On some partition functions, Amer. J. Math , P. Hagis, A problem on partitions with a prime modulus p 3, Trans. Amer. Math. Soc , P. Hagis, Partitions into odd summands, Amer. J. Math , P. Hagis, On a class of partitions with distinct summands, Trans. Amer. Math. Soc , P. Hagis, Partitions into odd and unequal parts, Amer. J. Math , P. Hagis, On the partitions of an integer into distinct odd summands, Amer. J. Math , P. Hagis, Some theorems concerning partitions into odd summands, Amer. J. Math , P. Hagis, Partitions with a restriction on the multiplicity of the summands, Trans. Amer. Math. Soc , P. Hagis, Partitions into unequal parts satisfying certain congruence conditions, J. Number Theory , G. H. Hardy and S. Ramanujan, Asymptotic formulae in combinatory analysis, Proc. London Math. Soc , L. K. Hua On the number of partitions of a number into unequal parts, Trans. Amer. Math. Soc , S. Isei, A partition function with some congruence condition, Amer. J. Math , S. Isei, On some partition functions, J. Math. Soc. Japan , S. Isei, Partitions in certain arithmetic progressions, Amer. J. Math , H. Iwaniec and E. Kowalsi, Analytic number theory, American Mathematical Society, J. Lehner, A partition function connected with the modulus five, Due Math. J , J. Livingood, A partition function with the prime modulus P > 3, Amer. J. Math , I. Niven, On a certain partition function, Amer. J. Math , A. V. M. Prasad and V. V. S. Sastri, H.R.R. series for certain -partitions using ord circles, Ranchi Univ. Math. J , H. Rademacher, On the partition function pn, Proc. London Math. Soc ,

19 A Hardy-Ramanujan-Rademacher-type formula for r, s-regular partitions V. V. S. Sastri, Partitions with congruence conditions, J. Indian Math. Soc. N.S , V. V. S. Sastri and S. Vangipuram, A problem on partitions with congruence conditions, Indian J. Math , no. 1-3, A. Sills, Towards an automation of the circle method, Gems in experimental mathematics, , Contemp. Math., 517, Amer. Math. Soc., Providence, RI, A. Sills, Rademacher-type formulas for restricted partition and overpartition functions, Ramanujan J , no. 1-3, A. Sills, A Rademacher type formula for partitions and overpartitions, Int. J. Math. Math. Sci. 2010, Art. ID , 21 pp. 29. G. B. Watson, A treatise on the theory of Bessel functions, 2nd ed., Cambridge University Press, 1944.