COMPUTING THE INTEGER PARTITION FUNCTION

Transcription

1 MATHEMATICS OF COMPUTATION Volume 00, Number 0, Pages S (XX) COMPUTING THE INTEGER PARTITION FUNCTION NEIL CALKIN, JIMENA DAVIS, KEVIN JAMES, ELIZABETH PEREZ, AND CHARLES SWANNACK Abstract. In this paper we discuss efficient algorithms for computing the values of the partition function and implement these algorithms in order to conduct a numerical study of some conjectures related to the partition function. We present the distribution of p(n) for N 0 9 for primes up to 03 and small powers of and 3.. Introduction Here we discuss some open questions concerning the partition function and algorithms for efficiently computing p(n) for all n up to some bound N. We then present some computational evidence related to these conjectures. A partition of a natural number n is a non-increasing sequence of natural numbers whose sum is n. The number of such partitions of n is denoted p(n). For example the partitions of 4 are: 4, 3 +, +, + + and Thus, p(4) = 5. One way of studying the partition function is to study its generating function. Euler [8] proved the following formula concerning this generating function. Received by the editor July 0, Mathematics Subject Classification. Primary (05A7);Secondary (P8, P83). Key words and phrases. partition function, discrete fast Fourier transforms. The authors were partially supported by NSF grant DMS The third author was partially supported by NSF grant DMS c 997 American Mathematical Society

2 NEIL CALKIN, JIMENA DAVIS, KEVIN JAMES, ELIZABETH PEREZ, AND CHARLES SWANNACK () P(q) := n 0 () p(n)q n = n q n = + q + q + 3q 3 + 5q 4 +. Euler s pentagonal number theorem asserts further that n ( q n ) = n Z ( ) n q (3n +n)/, from which we immediately deduce the recurrence which we will refer to as Euler s algorithm for computing p(n), (3) p(n) = k ( ( ) (p k+ n ) ( k(3k + ) + p n )) k(3k ). In fact, a careful analysis of the generating function for p(n) leads one to the Hardy-Ramanujan asymptotic formula (4) p(n) 4n 3 eπ n 3 which was improved by Rademacher [] to an exact formula for p(n) (see chapter 5 of [8] for a nice treatment of the expansion of p(n)). One would hope that the presence of an exact formula for p(n) would lead to a good understanding of the partition function or at least to efficient algorithms for the computation of p(n). Indeed, if one desires the value of p(n) for a single value of n then the exact formula of Rademacher yields a very fast algorithm. However, if one wishes to compute p(n) for all n N then Euler s algorithm is much faster. This is because once one already knows p(), p(),..., p(n ), the Euler algorithm only requires n additions to compute p(n) while the Rademacher formula requires that one compute the sum of n values of some quite complicated functions. Indeed, many questions concerning the partition function remain open and it is still computationally difficult to compute the values of p(n) for all values of n less than some bound N when N is large. We now outline some open questions and conjectures concerning the partition function for which we would like to gather numerical evidence. One of the simplest questions that one could ask is the frequency with which p(n) takes on even or odd values. Parkin and Shanks [] studied this question and were lead to the following conjecture.

3 Conjecture.. As n, we have #{n X : p(n)} lim = X X. One can of course ask more generally about the distribution of p(n) modulo an arbitrary modulus. In this direction Newman [7] made the following conjecture. Conjecture.. If M is a positive integer, then in every residue class r modulo M there are infinitely many integers n for which 3 p(n) r (mod M). Clearly the presence of the Ramanujan-type congruences bears on this question. Ramanujan [3] proved that (5) p(5n + 4) 0 (mod 5), (6) p(7n + 5) 0 (mod 7), (7) p(n + 6) 0 (mod ) for all n N. The congruence (8) p( 3 3n + 37) 0 (mod 3) was discovered in the late 960 s by Atkin, O Brien and Swinnerton- Dyer (see [9], [0] and []). Recently, Ono (see [9] and [4]) and Ahlgren and Ono [7] have proved the existence of infinitely many other such congruences modulo any prime l 5 while Ahlgren and Boylan [3] have proved that there are no other congruences of this type which are as simple as those of Ramanujan. In light of the above mentioned congruences, one might expect that the distribution of p(n) would be slightly biased to the zero class modulo a given integer M and otherwise uniform. Let us be more precise. Definition.. Let M Z and 0 r M and define for any X R δ r (M, X) := #{0 n < X : p(n) r X (mod M)}. Then we have the following conjecture of Ahlgren and Ono [6] Conjecture.3. Let M Z, 0 r M and let δ r (M, X) be defined as above. Then,

4 4NEIL CALKIN, JIMENA DAVIS, KEVIN JAMES, ELIZABETH PEREZ, AND CHARLES SWANNACK () If 0 r < M, then there is a real number 0 < d r (M) < such that lim X δ r(m, X) = d r (M). () If s and M = s, then for every 0 i < s we have d i ( s ) = s. (3) If s and M = 3 s, then for every 0 i < 3 s we have d i (3 s ) = 3 s. (4) If there is a prime l 5 for which l M, then for every 0 r < M we have d r (M) M In the direction of Conjecture., the best known result is due to Serre [8] and to Ahlgren []. We know that #{n X : p(n)} X and X #{n X : p(n)} log X, which is far from an affirmation of Conjecture.. In the direction of Newman s Conjecture., Atkin [9], Kolberg [5], Newman[7] and Klove [6] proved the conjecture for M =, 5, 7, 3, 7, 9, 9 and 3. Some conditional results were obtained in work of Ono [9], Ahlgren [], and Bruinier and Ono []. Recently, Ahlgren and Boylan [3, 4] have shown that the conjecture is true for M = l j for all primes l 5 and j. Like Conjecture., Conjecture.3 is still wide open. Part () of this conjecture is not known for any values of r and M. Theorem of [6] implies that if M is coprime to 6 then lim inf X δ 0(M, X) > 0. This is not known for any other (r, M) pairs. Thus it is would be of interest to gather numerical evidence on all of these conjectures. Additionally, the authors of [6] have shown that the congruences of p(n) are far more widespread than previously known. In particular, [6] has shown that for every prime l 5 and any positive integer m, there exist infinitely many arithmetic progressions of the form p(an + B) 0 (mod l m )

5 for every n Z. To be more precise, for each prime l 5, define the two integers ǫ l and δ l to be ( ) 6 ǫ l = l and δ l = l 4. Further, let S l be the set of (l + )/ integers S l = { β {0,,..., l } : ( β + δl Then, we have the following theorem from [7]. l ) = 0 or ǫ l }. Theorem.. If l 5 is prime, m is a positive integer, and β S l, then there are infinitely many non-nested arithmetic progressions {An + B} {ln + β} such that for every integer n. p(an + B) 0 (mod l m ) This naturally leads the authors of [7] to the following speculation [6]. Speculation.. If l 5 is prime and 0 r < l, define δ r(l, X) by δ r (l, X) = # {n < X : p(n) r (mod l) and n (mod l) S l} # {n < X : n (mod l) S l } is it true that lim X δ r (l, X) = l? It is of additional interest to investigate this speculation numerically. In the remainder of this paper we discuss a new algorithm for computing the values of the partition function and discuss its running time. We compare the running time of this new algorithm with that of Euler s algorithm. We then discuss a parallelization of the Euler algorithm for computing p(n) modulo a small prime p. We also discuss the scalability of the Euler algorithm. Finally, we present data related to the various conjectures mentioned above.. FFT Inversion of Power Series. To the authors knowledge the algorithm discussed in this section is new: it is exactly analogous to Euler s proof that the generating functions for partitions into odd parts and for partitions into even parts are identical. For any function f(z), the function f(z)f( z) is even. Hence, if f(z) = a n z n is a power series and if we write f(z)f( z) = b n z n, 5

6 6NEIL CALKIN, JIMENA DAVIS, KEVIN JAMES, ELIZABETH PEREZ, AND CHARLES SWANNACK then b n is nonzero only if n is even. This leads us to the following expansion for f(z). (9) f(z) = f( z) f(z)f( z) = f( z) f (z ), where f (z) = b n z n. Now, again f (z)f ( z) has only even coefficients and thus in the power series expansion of f (z )f ( z ) the only nonzero coefficients are the coefficients of z n where 4 n. Thus, we have (0) f(z) = f( z) f (z ) = f( z)f ( z ) f (z )f ( z ) = f( z)f ( z ). f (z 4 ) Repeating the above process k times yields () where () f(z) = f 0( z)f ( z )...f k ( z k ), f k+ (z k+ ) f 0 (z) = f(z) and f i (z ) = f i (z)f i ( z) i > 0. Thus we have that (3) f(z) = f 0( z)f ( z )...f k ( z k ) + O(z k+ ). Thus if one would like to compute the first N coefficients of f(z), it suffices to compute the first N coefficients of the product f 0 ( z)f ( z )...f k ( z k ) where k = lg N. One should also note that since we are interested only in the first N coefficients of, only the first f(z) N/i coefficients of f i are needed for this computation. In fact, we can say more precisely for any 0 l k that only the first N/ k l coefficients of each of the partial products are needed in this computation. f k ( z k )f k ( z k ) f k l ( z k l ) Definition.. Let R be any ring. We define the truncation operator T k : R[[z]] R[z] by k (4) T k ( a i z i ) = a i z i i=0 i=0

7 7 Put f i = T N/ i (f i ) and define (5) Then g 0 (z) = f k (z) and g i (z) = T N k i ( gi ( z ) f k i (z) ) i k. (6) g k ( z) = f 0 ( z) f ( z )... f k ( z k ) + O(z N ) = f(z) + O(zN ). Theorem.. Let f(z) be a power series with integral coefficients and constant coefficient. Let p be any prime which is congruent to modulo lg N +. The computation of the first N coefficients of /f(z) modulo p requires at most O(N lg N) coefficient multiplications. Proof. It suffices to show that the computation of g k ( z) described above requires only O(N lg N) coefficient multiplications where all arithmetic is done in F p and the f i s and g k s are understood to be in F p [z]. There are two steps in this computation. First, one must compute f (z), f (z),..., f k (z) where k = lg N using (). Next one must iteratively construct the g i s (i=0,,...,k) using (5). We recall that deg( f i (z)) = N/ i and that using the discrete fast Fourier transform, the computation of the product of two polynomials of degree less than M modulo p where M < N requires O(M lg M) coefficient multiplications. Thus the the number of coefficient multiplications required for the computation of the f i s is (7) O ( k ) N lg(n i ) = O (N lg N). i i=0 Now we consider the computation of the g i s. From (5), we have that deg(g i ( z )) < N and from (), deg( f k i k i )(z) = N. Thus k i the number of coefficient multiplications required to compute the g i s is at most (8) O ( k ) N N lg( k i k i) = O (N lg N) i=0 and this completes the proof of the proposition.

8 8NEIL CALKIN, JIMENA DAVIS, KEVIN JAMES, ELIZABETH PEREZ, AND CHARLES SWANNACK 3. Computing p(n). Theorem 3.. Let p be any prime which is congruent to modulo lg N +. The computation of p(n) modulo p for all n N requires at most O(N lg N lg p lg lg p) machine multiplications. Proof. (9) P(q) = Combining () and (), we see that p(n)q n = n=0 n Z ( )n q (3n +n)/ The computation of the first N terms of the denominator clearly requires at most O( N) multiplications of integers which are easily bounded in absolute value by N. Using the discrete FFT, one can perform multiplication of integers of size N using at most O( N lg N) machine multiplications. Thus, the first N terms of the denominator can be computed using at most O(N lg N) machine multiplications. Now we may use our inversion algorithm which will require an additional O(N lg N) coefficient multiplications. Since, we are working modulo p, each coefficient may be taken to be between 0 and p. Thus using discrete FFT, the multiple of any two coefficients requires at most O(lg p lg lg p) machine multiplications. Thus the first N coefficients of P(q) modulo p requires at most O(N lg N + N lg N lg p lg lg p) = O(N lg N lg p lg lg p) machine multiplications. Corollary 3.. The computation of p(n) for all n N can be done using O(N 3/ log (N)) machine multiplications. Proof. Select a prime p satisfying p(n) < p < (4Np(N)) 5.5 and p (mod lg N + ). To see that this can be done we note that the main result of [4] guarantees that for (A, B) =, there is a prime congruent to A modulo B that is smaller than B 5.5. Taking A = and B = lg N + p(n) then guarantees us that there is a prime p as above since lg N + < 4N. By Theorem 3. one can compute p(n) modulo p for all n N using O(N lg N lg p lg lg p) machine multiplications. However, since 0 < p(n) < p this gives the exact value of p(n) for all n N. Using (4), one can see that for N sufficiently large, we may assume that p < e 5 N. Combining this with our previous estimate on the number of machine multiplications yields the desired result. We will refer to the algorithm suggested by the previous two results as fast Fourier Transform inversion or simply as FFTI. A careful analysis of the Euler algorithm yields the following results. Theorem 3.. The computation of p(n) for all n N using the Euler algorithm requires O(N ) machine additions.

9 Proof. Examining (3), one sees that for each n N the computation of p(n) requires the addition of n / values of the partition function. Thus the computation of p(n) for all n N requires n 3/ additions of numbers which grow as large as p(n). The theorem now follows from (4) Theorem 3.3. For m N, the computation of p(n) modulo m for all n N requires at most O(N 3/ log m) machine additions. 9 Proof. The proof is the same as that of the previous theorem. The FFTI algorithm has a faster running time than the Euler algorithm for computing p(n) for all n N. However, if one is interested only in the values of p(n) modulo m where m is not a prime which is modulo lg N + and where log(m) < log (N) then the Euler algorithm has the faster running time. We now consider a parallel version of the Euler algorithm that improves the running time by a constant factor when the later case is of interest. 4. A parallel version of Euler s algorithm We now consider a parallel implementation of Euler s algorithm for computing p(n) for all n N. When computing in parallel one wishes to equally distribute the computation and data storage among all, say N p, processors. Since each partition number requires summing a subset of the previously computed numbers one can not store partition numbers linearly across the processors and expect balanced work across all N p processors at any given instance. Thus we store the partition number p(n) on the i th p processor if and only if n i p (mod N p ). Now, we make the following definitions. Definition 4.. Let m, n N 0 and m n, then the partial Euler sum of n up to m, p m (n), is p m (n) = k 0 n k(3k+)/ m j 0 n j(3j )/ m ( ( ) k+ p n ( ) j+ p ( n ) k(3k + ) + ) j(3j ).

10 0 NEIL CALKIN, JIMENA DAVIS, KEVIN JAMES, ELIZABETH PEREZ, AND CHARLES SWANNACK Definition 4.. Let i, m, n, N N 0, i < N and m n, then the i th N-wise decimation of the partial Euler sum of n up to m, q m,i,n (n), is ( ) q m,i,n (n) = ( ) k+ p n + k n k(3k+)/ i (mod N) 0 n k(3k+)/ m j n j(3j )/ i (mod N) 0 n j(3j )/ m ( ( ) j+ p n k(3k + ) ) j(3j ). Note that N i=0 q m,i,n(n) = p m (n) and p n (n) = p(n). Thus, to compute p(n) distributed across N p processors we can compute q n,ip,n p (n) on the i th p processor in parallel and upon completion, p n (n) on one processor. We now discuss how to efficiently compute the sums appearing in Definition 4.. In order to efficiently compute q m,i,n (n) we note that one does not need to check the condition n k(3k ±)/ i (mod N) for all values of k. By completing the square, we have n k(3k ±)/ i (mod N) if and only if (0) 6k + 4(n i) + (mod N) or () 6k 4(n i) + (mod N), provided that this square root exists. In this direction we provide the following definitions. Definition 4.3. Let i, N Z, N prime, N, 3. We define for any n Z, ( ) 4(n i) + χ i,n (n) = N ( ) where is the Legendre symbol. a p Definition 4.4. Let n, i, N N 0, N prime, N, 3, and χ i,n (n). Then, we define () t i,n (n) = 4(n i) + (mod N) where the square root in () is the unique integer r, 0 r < (N )/, such that r = 4(n i) + (mod N).

11 Thus, if n k(3k ± )/ i (mod N) then (3) 6k + t i,n (n) (mod N) or 6k t i,n (n) (mod N) Combining (3),(0) and () leads to the following definitions. Definition 4.5. Let n, i, N N 0, N prime, N, 3, and χ i,n (n). We define for any l Z, and for j {0,,, 3} α 0,i,N (l, n) = 6 ( t i,n (n)) + ln α,i,n (l, n) = 6 ( + t i,n (n)) + ln α,i,n (l, n) = 6 ( + t i,n (n)) + ln α 3,i,N (l, n) = 6 ( t i,n (n)) + ln ρ j,i,n (l, n) = n α j,i,n(l, n)(3α j,i,n (l, n) + ( ) j ) where 0 6 < N is the unique integer such that 6 6 (mod N). We will omit the subscripts i, N in Definitions 4.3 and 4.5 when they are apparent in the context they are used. Definition 4. may now be rewritten as (4) 3 ( ) αj(l,n)+ p (ρ j (l, n)) if χ(n) = q m,i,n (n) = j=0 j=0 l 0 ρ j (l,n) m l 0 ρ j (l,n) m ( ) α j(l,n)+ p (ρ j (l, n)) if χ(n) = 0 0 if χ(n) = Note that q m,i,n (n) vanishes whenever χ(n) =. Thus, since N p is a prime not or 3, χ ip,n p (n) for exactly (N p + )/ many 0 i p < N p for any given n N p. Thus, for any computation of p(n) only (N p + )/ processors are used. However, since N p is coprime to 6, (n i) runs through all equivalence classes mod N p for N p consecutive n. Thus, if q m,i,n (n) is computed for L consecutive values of n at any instance, where L = jn p and j N 0, each processor computes exactly j(n p + )/ non-zero q m,i,n (n). We now show how this can be used to compute p(n). Begin by letting n k be largest value for which p(n) is know exactly. On any processor, say i, we may then compute q nk,i,n(n k + ), q nk,i,n(n k + ),..., q nk,i,n(n k + L). Upon completion, one processor

12 NEIL CALKIN, JIMENA DAVIS, KEVIN JAMES, ELIZABETH PEREZ, AND CHARLES SWANNACK may then compute p nk (n k + ), p nk (n k + ),..., p nk (n k + L). Then, since p nk (n k + ) = p(n k + ) we may use this processor to compute p(n k + ), p(n k + ),..., p(n k + L). We call this one processor the control processor. This could be a separate processor in addition to the N p processors already in use. Our algorithm to compute p(n) is the following: Algorithm 4.. For any integer j > 0 and N p > 0, set L = jn p. On every processor except the control processor: (0) Set l = 0 () Have each processor compute p(n) exactly for 0 n < L () Set n k = L, l = (3) Compute q nk,i p,n p (n) on each processor, i p, using (4) for (l + )L n < (l + )L. (4) If l, receive exact values for p(n) for ll n < (l +)L and set n k = n k + L (5) Finish Computing q nk,i p,n p (n) on each processor i p using (4) for (l + )L n < (l + )L. (6) Send all computed q nk,i p,n p to control process (7) Set l = l + goto 3. On the control process we follow: (0) Set l =, n k = L () Receive q nk,i p,n p (n) from all processors for ll n < (l + )L. () Compute p nk (n) = N p i=0 q nk,i p,n p (n) for ll n < (l + )L (3) Set p(n k + ) = p nk (n k + ). (4) Compute p(n) exactly using p nk (n) and p(j) for n k j < n for ll n < (l + )L (5) Send exact p(n) to all processors (6) Set l = l +, n k = n k + L, goto 5. Discussion The algorithm of Section 4 was used with N p = 08 to compute p(n) modulo primes less than 04 for n 0 9. We list the statistical properties of the partition function up to 0 9 in the Appendix. Computations are ongoing and further data can be found at [5]. In order to be concise we only list the intermediate results for 0 6, 0 7, 0 8 and 0 9, and plot the cases in which we wish to be more precise. In regard to Conjecture. we see that the conjecture is justified. Examining Table we see that the distribution agrees out to the 4th decimal place. Similarly for M = 3, Table shows that the distribution is /3 out to 4 decimal places.

13 δ 0 (, X) δ (, X) Table : The Distribution of p(n) (mod ) δ 0 (3, X) δ (3, X) δ (3, X) Table : The Distribution of p(n) (mod 3) In order to examine the conjectures and speculation of [6] we make the following definitions. Definition 5.. Let M Z and define for any X Z and µ d (M, X) = M δ j (M, X) = δ 0(M, X) M M j= M σd (M, X) = (δ j (M, X) µ d (M, X)) M j= to be the mean and variance of the distribution of p(n) (mod M) among the non-zero congruence classes mod M for n < X. In regard to Conjecture.3 () the computational evidence suggests that this conjecture is justified. Examining Table 3 in the Appendix we can see that for primes M 03 it appears that the distribution of p(n) (mod M) in the zero class is approaching a limit, say d 0 (M). Additionally, by examining Table 4 in the Appendix we can see that the variance of the distribution δ j (M, X) is tending to zero, implying that the distribution of p(n) (mod M) approaches a limit. In fact, since the variance is tending to zero the computations suggest that p(n) is equally likely to lie in any of the non-zero classes modulo M. Thus, if (as the computations suggest) lim X δ 0 (M, X) = d 0 (M) and lim X σ d (M, X) = 0 then (5) lim X δ j(m, X) = d 0(M) M 0 < j < M.

14 4 NEIL CALKIN, JIMENA DAVIS, KEVIN JAMES, ELIZABETH PEREZ, AND CHARLES SWANNACK (5, X).5 (7, X).5 σ d σ d X (a) X (b) Figure. The variance of δ j for j not 0. For (a) M = 5 and (b) M = 7 and X from 000 to 0 9 in steps of 000. Table 4 also suggests that not only does the distribution of p(n) (mod M) converge, but it converges at a very fast rate. Indeed, examining Figure for M = 5 and M = 7 it appears that the variance is tending to 0 at an exponential rate. Further, examining Table 4 for all other primes this same trend appears. That is, as X grows linearly so does the exponent of the variance. This leads us to the following speculation. Speculation 5.. Is it true that for any M 3 and for any 0 j M log δ j (M, X) d j (M) lim > 0? X X As previously noted, due the congruence properties of p(n), it is reasonable to think that p(n) is biased toward the zero class. That is, d 0 (M) > d j (M) for 0 < j M. It is reasonable to expect that this holds for all prime powers M > 5 since, by Theorem., there are infinitely many non-nested arithmetic progressions p(an + B) 0 (mod M). However, if d 0 (M) > /M and (5) is true, this would imply that d j (M) < /M for 0 < j M. Examining Table 3 for small prime M > 5 it appears that indeed d 0 (M) > /M and thus Conjecture.3 (4) is also justified. However, since the known congruences modulo M for M > have A the influence of these progressions are not apparent in the computation for all primes. It is natural to consider how one may remove the bias of p(n) to the zero class. That is, it is natural to consider if one can, by excluding a subset of integers, say S, make the distribution of p(n) (mod M)

15 5 σ p(5, X) σ p(7, X) X (a) X (b) Figure. The variance of the the distribution of the congruence classes for p(n) where n S M. For (a) M = 5 and (b) M = 7 and X from 000 to 0 9 in steps of 000. uniform for n S. This is the content of Speculation.. Recall that for prime M 5, δ r (M, X) is defined to be the distribution of p(n) (mod M) for n (mod M) S M. That is, δ r(m, X) = # {n < X : p(n) r (mod M) and n (mod M) S M}. # {n < X : n (mod M) S M } Then, similar to Definition 5. we have the following definitions concerning the distribution of p(n) (mod M) for n S M. Definition 5.. Let M be a prime with M 5 and define for any X Z and µ p (M, X) = M σ p (M, X) = M M j=0 M j=0 δ j(m, X) = M ( δ j (M, X) µ p (M, X) ) to be the mean and variance of the distribution of p(n) (mod M) for n S M and n < X respectively. The computed values for σp(m, X) can be seen in Table 5 in the Appendix. The speculation of [6] seems to well justified. In fact the variance of the distribution of p(n) for n S M (mod M) again appears to decay exponentially in X. This can be seen in Figure (a) and (b) for M = 5 and M = 7. By examining Table 5 we can see that this trend appears for all computed primes. This leads us to the following speculation.

16 6 NEIL CALKIN, JIMENA DAVIS, KEVIN JAMES, ELIZABETH PEREZ, AND CHARLES SWANNACK Speculation 5.. Is it true that for any prime M 5 and for any 0 j M log δ j (M, X) /M lim > 0? X X A notable exception to the congruence properties of p(n) appear to occur for a modulus which is a power of or 3. That is, if Conjecture.3 () and (3) are true then p(n) is not biased toward the zero class modulo m or 3 m for any m N. In this direction we make the following definitions. Definition 5.3. Let M Z define for any X Z and µ(m, X) = M σ (M, X) = M M j=0 M j=0 δ j (M, X) = M ( δ j (M, X) ) M to be the mean and variance of the distribution of p(n) (mod M) for n < X respectively. The computed values for δ 0 (M, X) and σ (M, X) can be seen in Table 6 and Table 7 in the Appendix for powers of and the computed values for δ 0 (3 m, X) and σ (3 m, X) can be seen in Table 8 and Table 9 in the Appendix. Examining Table 6 and Table 8 we see that Conjecture.3 () and (3) is well justified. Indeed, for X = 0 9 there appears to be no bias toward the zero class as δ 0 agrees to the 5th decimal place. Further, in these cases the variance again appears to tend to zero exponentially fast further supporting Speculation 5.. In conclusion, we see that the computations suggest that Conjecture.3 and Speculation. seem to be well justified. In fact, our computations also suggested a stronger result than Conjecture.3. This leads us to the following revision of Conjecture.3. Conjecture 5.. Let M Z, 0 r M. Then, there exists a real number 0 < d(m) < such that and 0 < r < M In particular, lim δ 0(M, X) = d(m) X lim δ r(m, X) = d(m) X M

17 7 () If s and M = s, then d( s ) = / s () If s and M = 3 s, then d(3 s ) = /3 s (3) If there is a prime l 5 for which l M, then d(m) /M 6. Acknowledgments The authors would like to thank Dan Stanzione of the High Performance Computing Initiative, Fulton School of Engineering at Arizona State University and an anonymous reviewer for helpful comments. Appendix A. Data δ 0 (M, 0 6 ) δ 0 (M, 0 7 ) δ 0 (M, 0 8 ) δ 0 (M, 0 9 ) Table 3. The values of δ 0 for primes 03

18 8 NEIL CALKIN, JIMENA DAVIS, KEVIN JAMES, ELIZABETH PEREZ, AND CHARLES SWANNACK σd (M, 06 ) σd (M, 07 ) σd (M, 08 ) σd (M, 09 ) 3.83e e e e e-07.33e-08.75e e e-08.07e-08.36e-09.35e-.56e e e e e e e e e-08.8e e e e e e e e-08.33e e-0 5.3e e e e-0.974e e e e e e-08.59e-09.08e-0.678e- 4.89e-08.80e-09.3e-0.35e e e e-0.574e e-08.0e-09.56e-0.03e e e e-0.664e- 59.7e-08.80e e-0.5e e-08.60e-09.90e-0.504e e-08.83e-09.9e-0.7e- 7.86e-08.40e-09.45e-0.93e e-08.58e-09.98e-0.595e e-08.30e-09.36e-0.379e e e e-0.056e e e-0.039e-0.88e e e-0.93e-0.0e e e e-.4e- 03.e e e- 9.77e- Table 4. The variance of δ j (M, X) about µ d (M, X) for j 0

19 σp(m, 0 6 ) σp(m, 0 7 ) σp(m, 0 8 ) σp(m, 0 9 ) 5.6e e e e e e e-09.5e-0.3e e e-09.38e e-08.79e-08.40e e e e e-09.08e e e e-0.5e e e e-0.34e e e e e e e e-0 6.7e e e e e e e e e e e e e e e e e e e-09.94e e e e e-0.746e e e e e e e e-0.959e e e-09.67e-0.78e e-08.95e e-0.37e e e e-0.560e e-08.96e e-0.007e e-08.04e e-0.536e e-08.93e-09.6e-0.345e- 0.85e-08.0e e-0.960e e-08.47e e-0.804e- Table 5. The variance of δ j (M, X) about µ p(m, X) 9

20 0 NEIL CALKIN, JIMENA DAVIS, KEVIN JAMES, ELIZABETH PEREZ, AND CHARLES SWANNACK δ 0 (M, 0 7 ) δ 0 (M, 0 8 ) δ 0 (M, 0 9 ) /M Table 6. The values of δ 0 for powers of σ (M, 0 7 ) σ (M, 0 8 ) σ (M, 0 9 ) e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e-.4673e-.0085e e e e e e e-4 Table 7. The variance of δ j (M, X) about /M for powers of.

21 δ 0 (M, 0 7 ) δ 0 (M, 0 8 ) δ 0 (M, 0 9 ) /M Table 8. The values of δ 0 for powers of 3 σ (M, 0 7 ) σ (M, 0 8 ) σ (M, 0 9 ) e e e e e e e e e e e e e e e e e e e e e e-.5945e e e e e e e e e e e e e e e e e-6 Table 9. The variance of δ j (M, X) about /M for powers of 3.

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23 [] H. Rademacher, Topics in analytic number theory, Die Grundlehren der mathematischen Wissenschaften, Band 69, Springer-Verlag, New York-Heidelberg, 973. [3] S. Ramunajan, Congruence properties of partitions, Proceedings of the London Mathematical Society, 9 (99) [4] R. L. Weaver, New congruences for the partition function, Ramanujan J., 5 (00), no., [5] N. Calkin, J. Davis, K. James, E. Perez, and C. Swannack, cited 006: Statistics of the Integer Partition Function. [Available online at Neil Calkin, Department of Mathematical Sciences, Clemson University, Clemson, SC , USA address: calkin@clemson.edu Jimena Davis, Department of Mathematics, North Carolina State University, Raleigh, North Carolina 7695, USA address: jldavis9@unity.ncsu.edu Kevin James, Department of Mathematical Sciences, Clemson University, Clemson, SC , USA address: kevja@clemson.edu Elizabeth Perez, Applied Mathematics and Statistics, The Johns Hopkins University, G.W.C. Whiting School of Engineering, 30 Whitehead Hall, 3400 North Charles Street, Baltimore, MD 8-68, USA address: eaperez@ams.jhu.edu Charles Swannack, Department of Electrical and Computer Engineering, Clemson University, Clemson, SC 9634, USA Current address: Charles Swannack, Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, MA 039, USA address: swannack@mit.edu 3