A REMARK ON PRIME DIVISORS OF PARTITION FUNCTIONS

Transcription

1 International Journal of Nuber Theory c World Scientific Publishing Copany REMRK ON PRIME DIVISORS OF PRTITION FUNCTIONS PUL POLLCK Matheatics Departent, University of Georgia, Boyd Graduate Studies Research Center thens, Georgia 30602, US pollack@uga.edu Received (Day Month Year) ccepted (Day Month Year) Schinzel showed that the set of pries that divide soe value of the classical partition function is infinite. For a wide class of sets, weproveananalogousresultforthe function p (n) thatcountspartitionsofn into ters belonging to. Keywords: restrictedpartitionfunction;batean Erdősconjecture;Erdős Ivićconjecture; linear fors in logariths. Matheatics Subject Classification 200: P82, P83, N56. Introduction Let p(n) denote the classical partition function, defined as the nuber of ways of writing n as a su of positive integers, where the order of the suands is not taken into account. Hardy and Raanujan [0] developed the circle ethod in order to obtain precise estiates of the asyptotic behavior of p(n) as n!.theirresults were later refined by Radeacher [3], who found an exact expression for p(n) as the su of a rapidly converging series. Taking the first ter in Radeacher s series results in the stunning asyptotic forula p(n) = e p 2/3 p n 24 4 p 3(n 24 ) p 3/2/ q n 24 + O exp 2 p 2/3 p n. (.) In addition to its asyptotic properties, the arithetic properties of the partition function p(n) have drawn the attention of a nuber of authors. Once again, the story begins with Raanujan, who discovered the rearkable congruences p(5n + 4) 0 (od 5), p(7n + 5) 0 (od 7), and p(n + 6) 0 (od ), valid for all nonnegative integers n. s a weak consequence of this result, 5, 7, and each divide infinitely any values of p(n). Seventy years later, Schinzel showed that infinitely any pries divide soe eber of the sequence {p(n)} n=0. (See [8, Lea 2.]; see also [4] for quantitative results.) Schinzel s theore was

2 2 Paul Pollack superseded by work of Ono [2], who showed that every prie divides soe value of p(n). Very roughly speaking, Ono uses the theory of odular fors to show that congruences of the sort discovered by Raanujan are surprisingly ubiquitous. useful survey of related work is given in []. Despite these recent developents, Schinzel s ethod still has soe life to it. Quite recently, Cilleruelo and Luca [7] used a sophisticated variant of Schinzel s arguent to prove that the largest prie factor of p(n) exceeds log log n for (asyptotically) alost all n (see also the weaker result []). The purpose of this note is to establish the analogue of Schinzel s original theore for a wide class of restricted partition functions. If is a set of positive integers, we write p (n) for the nuber of partitions of n into parts all of which belong to. Equivalently, p (n) isdefinedbythe generating function identity P n=0 p (n)z n = Q a2 ( za ). Theore.. Let be an infinite set of positive integers. Suppose that satisfies the following hypothesis, referred to hereafter as condition P : There is no prie dividing all su ciently large eleents of. (P ) Then the set of pries that divide soe nonzero value of p (n) is infinite. For exaple, Theore. applies if is taken to be the set of perfect rth powers or the set of rth powers of pries, for any r. syptotics for these partition functions were studied by Hardy and Raanujan (see [9, 5]). It sees plausible to conjecture that the conclusion of Theore. in fact holds for every infinite set. Theore. does not apply to finite sets, but for these sets the situation is uch sipler. Let be a k-eleent set with k 2. There is a degree k polynoial f(z) 2 Q[z] with the property that p (n Q a2 a)=f(n) for every natural nuber n. (See, for exaple, [2].) For all but finitely any pries p, the coe cients of f are p-integral, and so f can be reduced od p. n eleentary variant of Euclid s proof of the infinitude of pries, dating back at least to Schur [5, pp. 40 4], shows that the reduction of f has a root od p for infinitely any pries p. In fact, the (non-eleentary) Frobenius density theore shows that f has a root od p for a positive proportion of all pries p. Thus, infinitely any pries in fact, a positive proportion of all pries divide a nonzero value of p (n). t present, Theore. appears to be one of only a handful of theores in the literature concerning arithetic properties of the partition functions p (n). Other exaples, concerned with the parity of p (n), can be found in the papers of Berndt, Yee, and Zaharescu [5,6]. 2. syptotic results for p (n) On the face of it, Schinzel s arguent depends crucially on the existence of the extreely precise asyptotic relation (.). In particular, it is iportant that the relative error there is of size O K (n K ) for each fixed K. Except for very special sets

3 Prie divisors of partition functions 3, this sharp of a result for p (n) is unavailable, and so certain auxiliary estiates ust be developed in order to get Schinzel s approach o the ground. We collect the needed results in this section. Throughout, all iplied constants ay depend on the choice of without further ention. We need soe notation. Let r denote the backwards-di erence operator, defined by rf(n) =f(n) f(n ). For each nonnegative integer k, weletr (k) denote the kth iterate of r. Welet (n) =r(k) p (n). We have the foral identity X X (n)zn =( z) k p (n)z n. n=0 The following estiates are contained in ore general results of Batean and Erdős (see [3, Theore 5]). Proposition 2.. Let be an infinite set of positive integers. Suppose that satisfies condition P. Then for each fixed nonnegative integer k, Moreover, as n!, n=0 (n)! as n!. p (k+) (n) (n)! 0. (2.) Batean and Erdős conjectured [3, p. 2] that (2.) could be sharpened to p (k+) (n) (n) k n /2 (2.2) for all large enough n. In fact, for each fixed k, they conjectured that (2.2) holds as long as satisfies a condition called P k : There are ore than k eleents of, and if we reove an arbitrary subset of k eleents fro, then the reaining eleents have gcd. (P k ) We are assuing that is infinite, and so our condition P guarantees that all of the conditions P k hold at once. Thus, the Batean Erdős conjecture iplies that under condition P, (2.2) holds for every k and all n>n 0 (k). The conjecture of Batean and Erdős has since been proved by Bell [4]. Writing (n)/p (j) (n), we obtain the following siple conse- (n) =p (n) Q k quence of Bell s theore. p (j+) Proposition 2.2. Let be an infinite set of positive integers satisfying condition P. Then for every fixed nonnegative integer k, and all large n, we have (n) k p (n)n k/2. (2.3) For our proof of Theore., it is necessary to have an upper bound on the successive di erences of log p (n). We bootstrap our way there, taking (2.3) as

4 4 Paul Pollack our starting point. s an interediate step, we study the successive di erences of p (n) j. For the reainder of this section, we assue that is infinite and satisfies condition P. Lea 2.3. Let j and k be fixed positive integers. For all large n, r (k) p (n) j j,k p (n) j n k/2. Proof. The proof is by induction, based on the product rule for r: r(f g)(n) =f(n )rg(n)+g(n)rf(n). (2.4) We call a foral expression in n a partition onoial if it is a product of ters of the for p (`) (n ), where ` and are nonnegative integers. The nuber of ters in the product will be called the degree of the onoial, while the su of the values of ` will be called the weight. For exaple, p (n 2) p (n 3) p (2) (n) p() (n ) has degree 4 and weight = 3. pplying r to a onoial yields a su of onoials of the sae degree and one higher weight. (This follows fro (2.4), by induction on the degree.) pplying this k ties, r (k) p (n) j can be written as a su of onoials of degree j and weight k. Now we estiate the size of an arbitrary onoial, viewed as a function of n.fix nonnegative integers ` and. Fro (2.3), p (`) (n ) p (n )n `/2 for large n. Fro (2.) (with k = 0), we have p (n ) p (n) as n!, and iterating this shows that p (n ) p (n). Hence, p (`) (n ) p (n)n `/2.Multiplying inequalities of this type together, we deduce that a fixed partition onoial of weight k and degree j is O j,k (p (n) j n k/2 ) for large n. Sincer (k) p (n) j is a su of O j,k () such onoials, the lea follows. We can now prove the needed result about the successive di erences of log p (n). Lea 2.4. Let k be a fixed positive integer. Then for all large n, we have Proof. We start by writing r (k) log p (n) = r (k) log p (n) k n k/2. = k ( ) log p (n ) (2.5) k ( ) log p (n ). (2.6) p (n)

5 Prie divisors of partition functions 5 Let 0 apple apple k. Repeated application of the k = case of (2.3) shows that p (n) p (n ) k p (n)n /2 for large n. Consequently, log p (n ) p (n ) = log + p (n) p (n) ( )` ` p (n ) = + O k (n k/2 ). ` p (n) `= If we substitute this back into (2.6), the accuulated error ter is O k (n k/2 ). To estiate the ain ter, write `= ( )` p (n ) ` p (n) ` = k p (n ) p (n) k + k 2 k 2 p (n ) + + 0, p (n) where 0,..., k are rational nubers depending only on k. Using this in (2.6), we obtain a ain ter of k k X j ( ) p (n ) j p (n) = j p (n) j = j p (n) j r (k) p (n) j. k ( ) p (n ) j The j = 0 ter contributes nothing to the final su, while by Lea 2.3, the ters j =,...,k each contribute O k (n k/2 ). The lea follows. 3. Proof of Theore. The following result, which is a consequence of Baker s estiates for linear fors in logariths, is due to Tijdean [6]. Proposition 3.. Fix a positive integer K. There is a constant C = C(K) with the following property: If N,M > 3 are integers both supported on the first K pries, with N>M, then N M>M/(log M) C. Proof of Theore.. Proposition 2. shows that p (n)!as n!.so we ay fix an n 0 with the property that p (n) > 0 once n>n 0. Let us suppose for the sake of contradiction that no ter of the sequence {p (n)} n=n 0 is divisible by a prie larger than the Kth prie, where K is a fixed positive integer. Let C = C(K)

6 6 Paul Pollack be as in Theore C, and let k denote the sallest positive integer exceeding C. For the rest of this proof, all iplied constants ay depend on, K, and k. Observe that there are arbitrarily large n for which r (k) log p (n) 6= 0. Otherwise, log p (n) eventually coincides with a polynoial in n. Sincep (n)!as n!, that polynoial ust be nonconstant with a positive leading coe cient. However, fro (.), log p (n) apple log p(n) p n. This is a contradiction. Fro now on, we assue n is chosen so that r (k) log p (n) 6= 0. Fro the expression (2.5) for r (k) log p (n), we see that exp(r (k) log p (n)) = B, where := Y 0appleapplek even p (n ) ( k ), B := Y 0appleapplek odd p (n ) ( k ). Let N = ax{, B} and M =in{, B}. ThenN>Mand, assuing n is large enough, both N and M are supported on the first K pries. By Lea 2.4, we have r (k) log p (n) n k/2 for large n. It follows that both /B and B/ are of the for + O(n k/2 ), and this shows that N M N n k/2. (3.) Both N and M are products of 2 k ters fro the set {p (n),...,p (n k)}. Since p (n) p (n k) as n!, we get that N M p (n) 2k. Consequently, So fro (3.), n /2 log p(n) log p (n) log M. N M M/(log M) k. But k>c. Thus, for n su contradicts Proposition 3.. ciently large (so that N and M are also large), this cknowledgents The author thanks Enrique Treviño and the referee for suggestions leading to iproveents in the exposition. References [] S. hlgren and K. Ono, Congruences and conjectures for the partition function, in q-series with applications to cobinatorics, nuber theory, and physics (Urbana, IL, 2000), Contep. Math., Vol. 29 (er. Math. Soc., 200), pp. 0. [2] G. lon and P. L. Clark, On the nuber of representations of an integer by a linear for, J. Integer Seq. 8(5) (2005), rticle , 7 pp. (electronic). [3] P. T. Batean and P. Erdős, Monotonicity of partition functions, Matheatika 3 (956) 4.

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