THE DISTRIBUTION OF THE NUMBER OF SUBGROUPS OF THE MULTIPLICATIVE GROUP

Transcription

1 THE DISTRIBUTION OF THE NUMBER OF SUBGROUPS OF THE MULTIPLICATIVE GROUP GREG MARTIN AND LEE TROUPE Abstract. Let In denote the number of isomorphism classes of subgroups of Z/nZ, and let Gn denote the number of subgroups of Z/nZ counted as sets not up to isomorphism. We prove that both log Gn and log In satisfy Erdős Kac laws, in that suitable normalizations of them are normally distributed in the limit. Of note is that log Gn is not an additive function but is closely related to the sum of squares of additive functions. We also establish the orders of magnitude of the maximal orders of log Gn and log In.. Introduction The distribution of values of additive functions has long been of interest to number theorists. Perhaps the most famous result in this area is the celebrated Erdős Kac theorem: if ωn and Ωn denote, respectively, the number of distinct prime factors of n and the number of prime factors of n counted with multiplicity, then the distributions of the values of both ωn log log n log log n and Ωn log log n log log n tend to the standard normal distribution. In other words, both ωn and Ωn are, in the limit, normally distributed with mean log log n and variance log log n. Indeed, Erdős and Kac [4] established this property for a large class of additive functions, and many subsequent authors have widened even further the set of functions for which we know such Erdős Kac laws. In this paper, we establish Erdős Kac laws for two functions that count subgroups of a natural family of finite abelian groups, as we now describe. Let Z n Z/nZ denote the multiplicative group of units modulo n. Let Gn denote the number of subgroups of Z n, counted as sets rather than up to isomorphism, so that G8 5, for example. The function Gn is not a multiplicative function of n, but it does have the property that Gn p φn G pn as we shall see below, where G p n denotes the number of p-subgroups of Z n. One could perhaps say that Gn is a multiplicative function of φn, or simply φ-multiplicative, making log Gn log G p n p φn a φ-additive function. Our primary aim is to show that log Gn possesses enough structure to satisfy a similar Erdős Kac law: 00 Mathematics Subject Classification. Primary N60, N45; secondary N37.

2 Theorem.. Define A 0 p log p 4 p 3 p + and p B 4 p and A A 0 + log p 3 p 4 p 3 p 3 p log p, p 6 p + p + p log and set C + A 3 0 log + 4A 0 + B Both sums are taken over all primes p. Then for every real number u, lim x x #{ n x: log Gn < Alog log n + u Clog log n 3/} u e t / dt. π In other words, the quantity log Gn is normally distributed, with mean Alog log n and variance Clog log n 3. We briefly indicate the overall structure of the proof of Theorem.. First, we understand the typical values of log Gn by writing them as a linear combination of squares of wellunderstood additive functions, together with one anomalous φ-additive function. Proposition.. Set X log log x / log log log x. For any positive integer n, define P n x log ωφn + ω q n Λq, 4 where Λq denotes the usual von Mangoldt function, and where the ω q are additive functions defined in Definition.7 below. Then for all but Ox/ log log log x integers n x, log log x 3/ log Gn P n x + O. 3 log log log x For any function fn, define the mean and set µf µf; x p x q X fp p, 4 Dx log µω φ + µω q Λq 5 4 so that Dx is simply P n x with each function of n replaced by its mean. Our strategy is to show that the values of P n x for n x are, asymptotically as x tends to, normally distributed with mean Dx and variance Clog log x 3, with C defined as in Theorem.. We carry out this strategy via the method of moments. Proposition.3. For any positive integer h, define the hth moment q X M h x Pn x Dx h. 6 Then lim x { M h x h! C h/ xlog log x 3h/ h/! h/, if h is even, 0, if h is odd.

3 h! The quantity for even h is precisely the hth moment of the standard normal distribution, and it is a famous lemma of Chebyshev that the normal distribution is determined h/! h/ by its moments see Section 7 for more details. Our proof is inspired by work of Granville and Soundararajan [8], who described a way to organize method-of-moments proofs in number theory to make the main terms more readily identifiable. The proof herein is tailored to the specific function P n x mentioned above, which can be viewed as a quadratic polynomial in increasingly many variables being evaluated at values of specific additive functions. For any fixed polynomial, one can apply the same techniques to its evaluation at values of additive functions from a much more general class, thereby obtaining Erdős Kac laws for these polynomials of additive functions as well including, for example, Erdős Kac laws for products of additive functions. This generalization is the subject of forthcoming work by the authors. Since Gn counts subgroups of Z n as sets, the reader might wonder about the equally natural function In that counts subgroups of Z n up to isomorphism. It turns out to be much easier to establish an Erdős Kac law for log In, partially because In is a φ- multiplicative function of a much simpler type, but mostly because we can leverage existing work of Erdős and Pomerance [6] on the number of prime factors of φn to greatly shorten our proof. Theorem.4. For every real number u, { lim x x # n x: log In < log log log n + u } log log log n3/ 3 π u e t / dt. In other words, the quantity log In is normally distributed, with mean log log log n and variance log log log 3 n3. The leading constant here, log , for the typical size of log In is a bit less than half the leading constant A for the typical size of log Gn in Theorem.; in other words, the total number Gn of subgroups of Z n is typically a bit more than the square of the number In of isomorphism classes of subgroups of Z n. We begin by establishing Proposition. in Section, which will require a brief digression into counting subgroups of finite abelian p-groups using partitions and Gaussian binomial coefficients. Sections 3 through 7 comprise the proof of Theorem., with the verification of Proposition.3 taking place in Section 6; a more detailed roadmap is provided in Section 3, along with notation and conventions that will be used through the rest of the paper. Finally, Section 8 contains the proof of the aforementioned theorem about In, along with proofs of the following maximal-order results for log Gn and log In: Theorem.5. The order of magnitude of the maximal order of log Gn is log x /log log x. More precisely, log x log x 6 log log x + O log log log x log x log x max log In log log x 4 log log x + O. log log x 3

4 Theorem.6. The order of magnitude of the maximal order of log In is log x/log log x. More precisely, log log x log x 5 log log x + O log x log x max log In π log log x 3 log log x + O. log log x. Expressing log Gn as a polynomial of additive functions In this section we prove Proposition.. First, we import a classical identity for the number of subgroups of a finite abelian p-group, which we alter into an approximate form that is suitable for our application. Then we describe exactly the p-sylow subgroup of the multiplicative group Z n and record its approximate number of subgroups. Finally we sum this contribution over all primes p, which mostly involves dealing with the complication of truncating this sum suitably to avoid being overwhelmed with error terms; we employ some anatomy of integers arguments to show that this truncation is valid for almost all integers n... Subgroups of p-groups. Let us recall, from the classification of finitely generated abelian groups, that every finite abelian group of size p m can be uniquely written in the form Z p α Z p α for some nonincreasing sequence α, α,... of nonnegative integers summing to m. We avoid naming the length of such sequences by the convention that all but finitely many of the α j equal 0. In other words, isomorphism classes of finite abelian p-groups are in one-to-one correspondence with partitions α α, α,... of m. A subpartition β of a partition α is a nonincreasing sequence β, β,... of positive integers such that β j α j for all j ; we write β α when β is a subpartition of α. It is easy though not quite trivial to see that Z p α Z p α contains an isomorphic copy of Z p β Z p β if and only if β α. We are interested in more precise information, however, about the number of subgroups of Z p α Z p α that are isomorphic to Z p β Z p β. Definition.. Given partitions β α and a prime p, define N p α, β to be the number of subgroups inside Z p α Z p α that are isomorphic to Z p β Z p β. Define N p α to be the number of subgroups inside Z p α Z p α as sets, not up to isomorphism, so that N p α β α N pα, β. As it happens, there is a classical formula for N p α, β, most conveniently expressed in terms of conjugate partitions. Every partition α has a conjugate partition a, which is most easily obtained by transposing the Ferrers diagram corresponding to α. The number of parts nonzero elements of the conjugate partition a is exactly equal to α, and in general α j equals the number of parts of a that are at least j in size; by the same token, the first part a of a is equal to the number of parts of α, and so on. We quote this classical formula, which can be found in [3, equation ] and the references cited therein: Lemma.. Let p be prime, and let β α be partitions. Let a a, a,..., a α, 0,... and b b, b,..., b β, 0,... be the conjugate partitions to α and β, respectively. Then α [ ] N p α, β p a j b j b aj j+ b j+ b j b j+ 4. p

5 Here, [ ] k l p otherwise defined by is the Gaussian binomial coefficient, defined to be 0 if l < 0 or l > k, and [ ] k l p l p k l+j. 7 p j The reader might gain some intuition from considering the case where α,...,, 0,... and β,...,, 0,... are the finest possible partitions of k and l, respectively, so that N p α, β is the number of l-dimensional subspaces of F k p. In this case, a k, 0,... and b l, 0,... and so N p α, β is simply [ ] k. It can be seen that the numerator of the l p formula 7 is, up to a power of p, the number of k l matrices over F p with full rank l and the column space of each such matrix defines an l-dimensional subspace of F k p, while the denominator is, up to the same power of p, the number of invertible l l matrices over F p which act by left multiplication on the set of k l matrices while preserving their column spaces. We will prefer an approximate version of the formula from Lemma., which the following pair of lemmas provides. Lemma.3. For any prime p and any integers 0 l k, there exists a real number 0 θ < 6 such that [ ] k p lk l + θp. l p Proof. For any integers k l j, we have the inequalities Note that and so l p k l pk l+j p j l p j j pk l+j p j l p j > pp, j p < + 6p, j p pp pk l. 8 p j where the last inequality follows by a simple calculation. Therefore equation 8 implies that l p lk l p k l+j l p lk l p j p < j plk l + 6p, which establishes the lemma. Lemma.4. Given partitions β α, let b and a be the partitions conjugate to β and α respectively. For any prime p, N p α, β for some real number 0 θ < 6. α p a j b j b j + θp α 5

6 Proof. By Lemmas. and.3, there exist real numbers 0 θ j < 6 such that α [ ] N p α, β p a j b j b aj j+ b j+ b j b j+ α p a j b j b j+ p b j b j+ a j b j + θ j p α p a j b j b j + θ j p. The lemma now follows from the intermediate value property of the continuous function fθ + θp α on the interval 0 θ 6, along with the observation that α f0 p a j b j b j α α p a j b j b j + θ j p f6 p a j b j b j. Finally, we want to sum N p α, β over all subpartitions β of α. It turns out that the dominant contribution to this sum comes from the subpartitions β nearest to α. Lemma.5. For any integer a 0 and any prime p, we have a p a bb p a /4+O and p a a a p a /4+O. b0 b0 Proof. Suppose first that a c is even. Using c bb c c b c c b for b c, we obtain a c c p a bb p c + p c bb p c + O p c c b p c + O p c, b0 which is certainly of the form p a /4+O. Even more simply, p a a a p c cc p a /4 exactly. Now suppose that a c + is odd. Using c + bb c + c + b c + c + b for b c, we obtain a c p a bb p cc+ + p c+ bb b0 p cc+ + O b0 c b0 p p c+ c+ b p cc+ + O p c+. b0 Since c + < cc +, the right-hand side is pcc+ p c+ 4, which is also of the form p a /4+O. On the other hand, p a a a p c+ cc p a /4+O as we have just seen. 6

7 Proposition.6. For any prime p and any partition α, log N p α log p 4 α a j + Oα log p. Proof. We recall our notation a a,..., a α, 0,... and b b,..., b α, 0,... for the conjugate partitions of α and β, respectively. Since N p α β α N pα, β by definition, Lemms.4 tells us that there exist constants 0 θ β < 6 and 0 θ < 6 such that N p α β α α p a j b j b j + θβ p α + θp α β α α p a j b j b j, 9 where the second equality again uses the intermediate value property of fθ + θp α and the positivity of each summand. On one hand, since every β α corresponds to certain choices 0 b j a j, we have β α α p a j b j b j a b 0 a α α b α 0 p a j b j b j α a j b j 0 α p a j b j b j p a j /4+O by Lemma.5. On the other hand, let β be the subpartition of α whose conjugate partition is b a,..., aα, 0,... ; considering only the summand on the right-hand side of equation 9 corresponding to β β yields β α α α p a j b j b j p a j a j a j α p a j /4+O by Lemma.5 again. Combining these last two inequalities with equation 9, we conclude that N p α + θp α α p a j /4+O, and therefore since log + x is bounded for 0 x < 3 α a j log N p α Oα O log p log p 4 as claimed. α a j + Oα log p.. Counting p-subgroups of the multiplicative group. We now begin the proof of Proposition. in earnest. As in the introduction, let Gn denote the number of subgroups of Z n and let G p n denotes the number of p-subgroups of Z n. Since every finite abelian group is the direct product of its p-sylow subgroups, it is easy to see that Gn G p n and thus log Gn log G p n. p φn Therefore, we first turn our attention to log G p n. It turns out that log G p n can be expressed in terms of arithmetic functions ω p jn, defined in two stages as follows: 7 p φn

8 Definition.7. For any positive integer q, let ω q n denote the number of distinct primes p n such that p mod q. For example, ω n ωn, while ω n ωn when n is even and ω n ωn when n is odd. These functions ω q will play a prominent role in the remainder of this paper. Already we start forming our intuition: since ωn is typically about log log n, and since one in every φq primes on average is congruent to mod q, the function ω q n is typically about φq log log n in size; and indeed, an Erdős Kac law for ω qn itself is straightforward to derive from the results in [4]. We must make a punctilious alteration to these functions ω q in order for them to exactly describe the structure of Z n. However, our intuition should also include the understanding that the difference between ω q and its sibling ω q defined momentarily is negligible in the distributional sense; in particular, all we will really use is that ω q n ω q n+o uniformly in integers n and prime powers q. Recall that the notation p r m means that p r m but p r+ m. Definition.8. For any prime power p r, define ω p rn +, if p is odd and p r+ n, ω p rn, if p is odd and p r+ n, ω n +, if p r and 3 n, ω p rn ω n +, if p r and n, ω n, if p r and n, ω rn +, if p and r > and p r+ n, ω rn, if p and r > and p r+ n. Definition.9. For any prime p and any positive integer n, let λ p n denote the largest power of p that divides the Carmichael function λn. In other words, λ p n is the exponent of the p-sylow subgroup of Z n. Lemma.0. Let n be a positive integer, and let p be a prime dividing φn. The p-sylow subgroup of Z n is isomorphic to Z p α Z p α, where α α, α,... is the conjugate partition to a a, a,... ω p n, ω p n,..., ω p λpnn, 0,.... Proof. First let p be an odd prime. Write the p-sylow subgroup of Z n as Z p α Z p α for some partition α which we want to determine. There are two possible sources of factors of p in φn: primes q n such that q mod p including those congruent to modulo higher powers of p, and p itself or a higher power of p dividing n. Furthermore, by the Chinese remainder theorem and the existence of primitive roots modulo every odd prime power, we can say exactly how each of these sources affects the p-sylow subgroup of Z n. Each prime q n such that q mod p j contributes, to the the p-sylow subgroup of Z n, a factor of Z p m with m j indeed, m is the exponent of p in the prime factorization of q. Moreover, if p j+ n, then this power of p contributes to the p-sylow subgroup of Z n another factor of Z p m with m j in this case, m + is the exponent of p in the prime factorization of n itself. All factors of the form Z p m in the primary decomposition of Z n arise in one of these two ways; therefore, the number of factors of order at least p j in the 8

9 p-sylow subgroup of Z n is exactly equal to ω p jn. But a j, the jth entry in the conjugate partition to α, is precisely the number of factors of order at least p j in Z p α Z p α. We conclude that a j ω p jn as desired. The case p follows by an similar analysis, complicated slightly by the fact that Z Z and Z 4 Z while Z Z r r Z when r 3. It is worth remarking that in particular, Lemma.0 shows that the exponent of p in the prime factorization of φn is exactly λ pn ω p jn for every prime p. Consequently, p φn λ pn ω p jn log p log φn. 0 Furthermore, let ν p n denote the power of p in the prime factorization of n. The proof of Lemma.0 also shows that for odd primes p, λ p n max { ν p n, max{j : ω p jn } } ; when p, we must replace ν p n with max{0, ν n }. In either case, { λ p n max ν p n, } ω p jn. j With the following proposition, we may leave most of the details of abelian groups and partitions behind and operate within the realm of analytic number theory to complete the proof of Proposition.. Proposition.. For any positive integer n and any prime p dividing φn, log G p n log p 4 λ pn Moreover, if p φn, then log G p n log. ω p jn + Oλ p n log p. Proof. If p φn, then the p-part of Z n is precisely Z p, which trivially contains exactly two subgroups; hence G p n in this case. In general, Proposition.6 tells us that log N p α log p 4 α a j + Oα log p, while Lemma.0 gives us the exact evaluations α λ p n and a j ω p jn..3. Counting all subgroups of the multiplicative group. The main goal of this section is to establish Proposition., which says that log Gn is approximately equal to a particular polynomial expression in additive functions of n, at least for most integers n. Several times in the course of these proofs, we will make use of upper bounds of the correct order of magnitude that follow, via partial summation, from the prime number theorem, or indeed from Mertens s formulas or even Chebyshev s bounds for prime-counting functions. Such sums include sums over primes like p y /p or p>y /p, or sums over prime powers like p j y log p j /p j or q y Λq/q. Moreover, since q/φq for all prime powers q, such sums can also be modified to have denominators of p instead of p, or φq instead 9

10 of q. In all such cases, we shall simply say by partial summation to indicate that the required upper bounds follows in a standard way from these prime-counting estimates. In addition, we will make frequent use of the following Mertens-type estimate for arithmetic progressions, which can be found in [] or []: Lemma.. For q x, we have For the rest of this section, we set p x p mod q W log log log x p log log x φq X log log x / log log log x Y log log x. Note that this definition of X is the same as in Proposition.. Lemma.3. For all but Ox/W integers n x, { max log p, λ p n log p, ν p n log p, p Y p Y p Y λ pn p j Y log q + O φq } ω p jn log p. log log x log log log x. Proof. The first sum on the left-hand side of equation is clearly bounded above by the second sum; and this second sum, by equation, is bounded above by the maximum of third and fourth sums on the left-hand side. It therefore suffices to show that ν p n log p + ω p jn log p x log log x log Y, 3 p Y p j Y for then there can be no more than Ox/W integers n x for which either of the two summands exceeds log log x log Y W log log x log log log x. The first sum on the left-hand side of equation 3 can be bounded simply: ν p n log p p Y p Y log p j p j n p Y p Y log p j log p j p j n x p j x p Y log p p x log Y by partial summation, which is more than sufficient. As for the second sum on the left-hand side of equation 3, ω p jn log p log p log p q x p j Y p j Y q n q mod p j 0 p j Y log p p j Y q mod p j q x q mod p j q n x q.

11 Since Y < log x when x is large enough, Lemma. yields ω p jn log p x log log x logp j log p + O φp j φp j p j Y p j Y x log log x p j Y log p φp j + x x log log x log Y + x log Y p j Y log p j φp j by partial summation, completing the verification of the bound 3. The following lemma is very similar to known results see [6] for example on the scarcity of numbers n for which φn is divisible by the square of a large prime. Lemma.4. All but Ox/W integers n x have both λ p n for all p > Y and ω p jn for all p j > Y. Proof. First, fix a prime p > Y. If λ p n, then either p 3 n or there exists a prime q n with q mod p ; the number of integers n x satisfying one of these two conditions is at most Y <p x /3 x p 3 + q x q mod p Therefore the total number of integers n x for which λ p n for even a single prime p > Y is, by Lemma., at most x p + x 3 q Y <p x q x q mod p < p>y x q. x p + x log log x log p + O 3 φp φp p>y log log x x Y log Y + x Y log Y + Y by partial summation; this is an acceptably small bound for the number of such n x, given our choices of Y and W. Similarly, fix a prime power p j > Y. If ω p jn, then there exist two distinct primes q and r dividing n such that q r mod p j. The number of integers n x satisfying this condition is, by Lemma., at most q<r x q r mod p j x qr < x q x q mod p j log log x x + φp j q log pj log log x x + log pj. φp j p j p j

12 Therefore the total number of integers n x for which ω p jn for even a single prime power p j > Y is log log x x + log pj log log x x + log Y x p j p j Y log Y Y W p j >Y again by partial summation. We now have collected enough results to obtain a not-quite-final version of Proposition. where, for the moment, the range of summation p j Y rather than p j X is longer than we would like. Lemma.5. For all but Ox/W integers n x, log Gn ωφn log + ω 4 p jn log p + Olog log x log log log x. Proof. By Proposition., log Gn log G p n p φn p φn λ pnω pn p φn λ pnω pn p φn log + log + O log + O p j Y p φn λ pnω pn p φn λ pnω pn p φn λ pnω pn Since λ p n for all p φn, log + λ p n log p p φn λ pnω pn Oλ p n log p + log p λ pn 4 λ p n log p + log + λ p n log p + p φn λ pnω pn p φn λ pnω pn j λ pn p φn λ pnω pn j λ pn λ p n log p. ω p jn ω 4 p jn Λp j ω 4 p jn Λp j. Furthermore, by Lemma.4, for all but Ox/W integers n x we never have λ p n for any p > Y ; for these non-exceptional integers, we can therefore incorporate the condition p Y into the relevant sums, yielding log Gn p φn log + O p Y λ p n log p + p Y p φn λ pnω pn j λ pn ω 4 p jn Λp j.

13 In this last sum, for all but Ox/W integers n x, Lemma.4 also implies that ω p jn and thus ω p jn for all p j > Y, which implies ω p jn Λp j Λp j p Y Λp j λ p n log p; p Y j λ pn p Y p φn λ pnω pn j λ pn p j >Y p Y p φn λ pnω pn j λ pn p j >Y therefore for these non-exceptional integers, we can strengthen the condition p Y to p j Y in the last sum to obtain log Gn log + O λ p n log p + ω 4 p jn Λp j p φn ωφn log + p Y p φn λ pnω pn j λ pn p j Y p φn λ pnω pn j λ pn p j Y ω 4 p jn Λp j + O log log x log log log x, 4 where the second equality is valid for all but Ox/W integers n x by Lemma.3. From the definition of ω p j, we know that ω p jn ω p jn + O, and consequently ω p jn ω p jn + O ω p jn +. In particular, p φn λ pnω pn j λ pn p j Y ω 4 p jn log p p φn λ pnω pn j λ pn p j Y p φn λ pnω pn j λ pn p j Y 4 ω p jn + Oω p jn + log p ω 4 p jn log p + O p j Y ω p jn log p + p Y λ p n log p, and by Lemma.3 this error term is also log log x log log log x for all but Ox/W integers n x. Therefore we may modify equation 4 to log Gn ωφn log + ω 4 p jn log p + Olog log x log log log x. p φn λ pnω pn j λ pn p j Y In this sum, we may remove the condition of summation λ p nω p n at a cost of at most p Y λ 4 pn log p, which again is negligible for all but Ox/W integers n x by Lemma.3. Since ω p jn 0 whenever p φn or j > λ p n, we may remove the conditions p φn and j λ p n as well. This establishes the lemma. Finally, we show that we can truncate the range of summation in the above lemma from p j Y down to p j X at the cost of a larger error term, thereby obtaining Proposition.. 3

14 Proof of Proposition.. In the notation of Proposition. and of this section, Lemma.5 states that for all but x/w integers n x, log Gn P n x + ω 4 q n Λq + Olog log x log log log x. X<q Y Therefore it suffices to show that for all but x/w integers n x, the sum in the equation above is log log x 3/ / log log log x. In turn, this statement can be established by showing that ω q n xlog log x3/ Λq log log log x. 5 We may write X<q Y X<q Y ω q n Λq X<q Y X<q Y X<q Y X<q Y Λq Λq Λq Λq p n p mod q p,p n p p mod q p p mod q p,p n p mod q For the first sum, Lemma. gives Λq X<q Y p n p mod q x X<q Y X<q Y Λq x log log x + p n X<q Y p x p mod q Λq Λq p n p x p mod q X<q Y Λq φq by partial summation. For the second sum, we argue similarly: Λq x Λq p p X<q Y p,p x p p n p p mod q X<q Y xlog log x q>x p,p p p mod q Λq q p p p mod q p p x log log x log Y xlog log x /X p p n. xlog log x3/ log log log x. These last two upper bounds establish the estimate 5 and therefore the proposition. 4

15 3. Notation and setup In this section, we prepare some notation we will need to prove Proposition.3. At the end of the section, we outline the main stages of the proof, which span the next several sections. Definition 3.. Define the function ω 0 n ωφn. Comparing with Definition.7 shows that this notation ω 0 is mathematically dubious, but it will be typographically convenient. For example, we note that ω q p log z for every q 0 and every prime p z: when q this is obvious from Definition.7, while for q 0 we have ω 0 p ωp logp / log. In this notation, the definitions and 5 become P n x log ω 0 n + ω q n Λq 4 q X Dx log µω µω q Λq, 4 where by equation 4 we may simply write µω q p x q X ω q p p for every q 0. We shall continue to write ranges of summation over q as either q, when the sum excludes q 0, or 0 q, when the sum includes q 0. By way of intuition, the typical size of ω 0 n is log log x ; this is quite a bit larger than the typical size of any ω q n with q but, on the other hand, these ω q typically occur squared, while the function ω 0 typically occurs to the first power. Consequently, the contribution of the two types of function to the typical size of P n x is of the same order of magnitude. The typical size of P n x, as n varies over integers up to x, is asymptotically Dx due essentially to linearity of expectation, and consequently the distribution of the difference P n x Dx will be the main focus of our investigation. Definition 3.. For any prime p, define the function { /p, if p a, f p a /p, if p a. We extend this function completely multiplicatively in the subscript not, as might be expected, in the argument: for any positive integer r, we set f r a p α r f p a α. Finally, we set F ωq a p x ω q pf p a. 5

16 Notice that, for any n x, we have the exact identity ω q n µω q + F ωq n. 7 We have thereby decomposed an additive function into its mean value on the integers up to x which is asymptotically equal to µω q and a term F ωq n that oscillates as n varies. This innovation, due to Granville and Soundararajan [8], allows for a more direct identification of the main terms that arise in the calculations of the hth moments M h x P n x Dx h of the difference between P n x and its mean value. We approach these moments by first rewriting P n x using equation 7: P n x log ω 0 n + ω q n Λq 4 q X log µω 0 + F ω0 n + 4 q X µω q + F ωq n Λq. Upon expanding the square inside the sum and then subtracting Dx, we obtain P n x Dx log F ω0 n + µω q F ωq n + F ωq n. 8 4 q X q X We then estimate the hth moment by taking the entire right-hand side to the hth power, expanding into a sum of 3 h terms, and estimating each term separately. The terms with the fewest F -factors will comprise the main term for M h x, while the others contribute only to the error term. The bookkeeping and notation involved with tracking all of these terms is quite messy, and we have organized the remainder of the paper as follows to minimize the trauma to the reader. The aim of Section 4 is to introduce a general algebraic framework for handling the terms that arise upon expanding the hth power of the right-hand side of equation 8. Section 5 is devoted to proving asymptotic estimates and formulas for those individual terms as x tends to infinity. In Section 6, we complete the proof of Proposition.3 using the results of previous sections. Finally, in Section 7, we quickly justify our use of probabilistic language in the statement of Theorem. by deducing Theorem. from Theorem Polynomial accounting The main goal of this section is to establish Proposition 4.7, which is used to identify and simplify the main term of the moments M h x for h even at the end of Section 6. The proof begins with some combinatorial arguments, concerning polynomials in many variables, which are elementary but extremely notation-intensive. Along the way, we also introduce some polynomial-related notation Definition 4.5 for future use. Definition 4.. For any positive integer k, define Σ k to be the set of all permutations of {,..., k} that is, the set of all bijections from {,..., k} to itself. A typical element of Σ k will be denoted by σ. 6

17 For any positive even integer k, define T k to be the set of all -to- functions from {,..., k} to {,..., k/}. A typical element of T k will be denoted by τ. We let τ 0 denote the orderpreserving element of T k defined by τ 0 j j for each j k. For τ T k and j {,..., k/}, define Υ j and Υ j to be the two distinct preimages of j in {,..., k}; we will never need to distinguish between the two. Lemma 4.. Let k be a positive even integer. The function ψ : Σ k T k defined by ψσ τ 0 σ is surjective and k/ -to-. Proof. Given any τ T k, the equality ψσ τ holds for a particular σ Σ k if and only if {Υ j, Υ j} {σj, σj} for every j k. 9 This specifies each of the k unordered pairs {σj, σj}, each of which provides a choice of two options for which element equals Υ j and which equals Υ j; the total number of preimages σ is thus exactly k/. The following definition and lemma provide one of our main tools for dealing with arbitrary powers of finite sums. Definition 4.3. Let h and l be positive integers with h even. Let R be a commutative ring of characteristic zero with a unit element, and define two commutative polynomial rings over R with l+ and l+ variables: let x 0,..., x l be indeterminates and define S R[x 0,..., x l ], and let {z ij : 0 i, j l} be indeterminates and define S R[z 00,..., z ll ]. Let S h be the R-submodule of S spanned by monomials of total degree h, and let S h/ be the R-submodule of S spanned by monomials of total degree h/. We define an R-module homomorphism Φ h : S h S h/ in the following way. Given a monic monomial M x m x mh in S h where m,..., m h {0,..., l} are not necessarily distinct, set Φ h M z mσ m h! σ z mσh m σh. σ Σ h Note that the order of the indices m,..., m h is not uniquely defined by M, but this is not problematic since the sum defining Φ h M averages over all permutations σ. Then we extend Φ h R-linearly to S h, so that Φ h j r jm j j r jφ h M j for any monic monomials M j S h and elements r j R. For example, with h 4 and l, Φ 4 x 0x x 7x 3 x 6 z z z z z 0z z 0z z 0z z 0z 0 7 z z 7 z z. Lemma 4.4. Let h and l be positive integers with h even. Let R be a commutative ring of characteristic zero with a unit element, and let Φ h be defined as in Definition 4.3. For any elements r 0,..., r l R, Φ h r0 x r l x l h h/ r i r j z ij. 7 0 i,j l

18 Proof. The key to the calculation is to purposefully avoid expanding r 0 x r l x l h using multinomial coefficients; allowing repetition such as x + x x + x x + x x + x makes the counting argument much easier. By the definition of Φ h, l h l Φ h r i x i Φ h i0 m 0 l l h! Since r m r mh r mσ r mσh Φ h l i0 m 0 h r i x i h! m h 0 l σ Σ h m 0 l m h 0 r m r mh x m x mh r m r mh h! l m h 0 σ Σ h z mσ m σ z mσh m σh r m r mh z mσ m σ z mσh m σh. for any σ Σ h, we can rewrite this identity as l σ Σ h m 0 l m h 0 r mσ r mσh z mσ m σ z mσh m σh. Now the only effect of any fixed σ on the inner h-fold sum is to permute the order of the indices; therefore setting j m σ, j m σ, and so on, we may write Φ h l i0 h r i x i h! σ Σ h j 0 l l r j r jh z j j z jh j h. j h 0 The inner h-fold sum no longer depends on σ, and so Φ h l i0 r i x i h l j 0 l l r j r jh z j j z jh j h j h 0 j 0 j 0 l h/ r j r j z j j, which is equivalent to the statement of the lemma. Remark. The map Φ h can also be interpreted as a rather natural R-module homomorphism from Sym h M to Sym h M R M, where M R l+ S. However, this interpretation does not seem to shorten the verification of the desired identity. We now wish to apply these results to a specific polynomial related to the moments of log Gn. Given a real number x, let X log log x / log log log x as before, and let ρx denote the number of prime powers up to X. Define the polynomial Qx 0, x,..., x ρx log x ρx i Λq i x i. Note that the polynomial P n x defined in the introduction is equal to this polynomial Q evaluated at the tuple x 0, x,..., x ρx ω 0 n, ω q n,..., ω qρx n. For consistency, we will abuse notation and set q 0 0; this will be convenient when applying the results of this section to P n x in Section 6. 8

19 Let Q i denote the partial derivative of Q with respect to x i. Observe that Qx 0 + y 0,..., x ρx + y ρx Qy 0,..., y ρx log x 0 + Definition 4.5. Let h be a positive integer. Define ρx i0 ρx i Λq i x i y i + x i Q i y 0,..., y ρx + ρx i ρx i Λq i x i Λq i x i. R h x 0,..., x ρx, y 0,..., y ρx Qx 0 + y 0,..., x ρx + y ρx Qy 0,..., y ρx h. To expand this out in gruesome detail, R h can be written as the sum of some number B h of monomials: B h R h x 0,..., x ρx, y 0,..., y ρx β k hβ k hβ r hβ x vh,β,i i 0 y wh,β,j, where each vh, β, i and wh, β, j is an integer in {0,,..., ρx}; the total x-degree of the βth monomial in the sum is k hβ, while its total y-degree is k hβ. From equation 0, we see that each k hβ is between h and h inclusive, each k hβ is at most h, and each k hβ + k hβ is also between h and h. As it turns out, the most significant monomials on the right-hand side of equation are those of minimal x-degree, that is, those monomials with k hβ h. These monomials will contribute to the main term of the calculation of the hth moment in Section 6 when h is even, while the other monomials contribute only to the error term. Consequently we focus on these special monomials for the remainder of this section. Lemma 4.6. The part of R h of total x-degree h is β B h k hβ h r hβ h i x vh,β,i k hβ y wh,β,j ρx i0 x i Q i y 0,..., y ρx h. Proof. The left-hand side is exactly the definition of the part of R h of total x-degree h, or equivalently since k hβ h always the part of R h of total x-degree at most h. But R h is the hth power of the polynomial Qx 0 + y 0,..., x ρx + y ρx Qy,..., y ρx, whose part of total x-degree at most equals ρx i0 x iq i y 0,..., y ρx by equation 0. We are now ready to establish the proposition that will be used in Section 6 when analyzing the main term of the even moments. For any positive even integer h, define s h h! h/ h/!. 3 9

20 Proposition 4.7. Let h be a positive even integer. In the notation of Definitions 4. and 4.5, h/! β B h k hβ h r hβ k hβ h/ y wh,β,j τ T h i ρx s h z vh,β,υ ivh,β,υ i i0 ρx j0 Q i y 0,..., y ρx Q j y 0,..., y ρx z ij h/. 4 Proof. Consider the operator Φ h from Definition 4.3, using the ring R R[y 0,..., y ρx ]. We establish the lemma by showing that the left- and right-hand sides of equation 4 are the results of applying Φ h to s h times the left- and right-hand sides, respectively, of equation. Checking the right-hand side is easy: since Φ h is an R-module homomorphism, ρx h ρx h Φ h s h x j Q j y,..., y ρx s h Φ h Q j y,..., y ρx x j j0 ρx s h i0 j0 ρx j0 Q i y 0,..., x ρx Q j y 0,..., y ρx z ij h/ by Lemma 4.4 with r j Q j y,..., y ρx R. As for the left-hand side: by R-linearity we have k h hβ Φ h s h r hβ y wh,β,j β B h k hβ h i x vh,β,i s h β B h k hβ h s h β B h k hβ h r hβ r hβ k hβ y wh,β,j Φ h h i k hβ y wh,β,j h! x vh,β,i σ Σ h z vh,β,σvh,β,σ z vh,β,σh vh,β,σh. But by Lemma 4., the set Σ h can be partitioned into subsets of size h/, each subset corresponding to a particular τ T K and consisting of those σ for which equation 9 holds. Therefore k h hβ Φ h s h r hβ y wh,β,j β B h k hβ h s h i β B h k hβ h r hβ x vh,β,i k hβ h/ y wh,β,j z vh,β,υ vh,β,υ h! z vh,β,υ h/vh,β,υ h/. τ T h 0

21 The lemma now follows upon noting that s h h/ /h! / h!. 5. Covariances of two additive functions The goal of this section is to evaluate certain expressions, arising from expanding the hth power of the right-hand side of equation 8, in terms of certain covariances which we now define. Throughout this section, k is a fixed positive integer, x > is a real number, and z x /k. For any two additive functions g and g, define their covariance to be covg, g covg, g ; z p z g pg p p p Whenever g p, g p log p as will be the case in our application, this definition can be simplified to covg, g g pg p log p + O g pg p + O. 5 p p p p z p z p z We begin by finding asymptotic formulas for these covariances when each of g and g is equal to one of the ω q q 0. The Bombieri Vinogradov theorem will be an essential tool here and later in the paper; see for example [9, Theorem 7.] for the statement for the function ψx; q, a, from which it is simple to derive the analogous versions for the functions θx; q, a and πx; q, a an example of such a derivation is the proof of [, Corollary.4]. Theorem 5.. For any positive real number A, there exists a positive real number B BA such that the estimates max a,q θx; q, a x φq x A 6 log x A q Q max lix a,q πx; q, a φq x A 7 log x A q Q hold for all x >, where Q x / log x B. The following three lemmas provide the desired evaluations of the relevant covariances; we must attend separately to the cases where neither, one, or both of the two additive functions equals ω 0. Lemma 5.. Let q and q be powers of primes possibly of the same prime, and let [q, q ] denote the least common multiple of q and q. Then covω q, ω q log log z φ[q, q ] + O. uniformly for q, q z.

22 Proof. Since each ω qi is uniformly bounded, and ω q pω q p precisely when p is congruent to modulo [q, q ], equation 5 becomes log log z covω q, ω q + O p φ[q, q ] + O log[q, q ] + O φ[q, q ] p z p mod [q, q ] by Lemma.; and the first error term can be absorbed into the O. Lemma 5.3. If q z /4 is a prime power, then covω q, ω 0 log log z φq + Olog log z. Proof. We emulate the proof of [6, Lemma.]. In preparation for a partial summation calculation, we first show that t log log t t ω q pω 0 p ωp φq log t + O 8 log t p t p t p mod q for all t > q; the first equality follows from Definition.7 of ω q and Definition 3. of ω 0. When q > log t this estimate is simple: the trivial bounds πt; q, < t/q and ωp log p result in p t p mod q ωp πt; q, log t < t log t q t log t, which is consistent with the right-hand side of equation 8 since q > log t implies that log log t/φq log log t log log log t/ log t. Consequently, we may assume that q log t. Letting l denote a variable of summation taking only prime values, and noting that at most primes greater than p /3 can divide p, ωp + p t p mod q p t p mod q l p l t /3 l t /3 p t p mod q p mod l l t /3 l q l t /3 l q p t p mod ql l p l>t /3 + O p t p mod q + O l t /3 l q p t p mod q πt; ql, + O ωqπt; q, l t /3 l q + Oπt; q, t πt; ql, + O, log t 9

23 where the last step follows from the Brun Titchmarsh theorem see [0, Theorem 3.9] and the assumption q log t: t ωqπt; q, ωq φq logt/q ωq t φq log t t log t. 30 By the Bombieri Vinogradov estimate 7 with A noting that every modulus ql in the sum is at most t /3 log t t / log t B, πt; ql, lit φql + O lit πt; ql, φql l t /3 l q l t /3 l q lit φq l t /3 l q l t /3 l q l + O t log t lit log log t /3 + Oωq t + O φq log t t log log t t φq log t + O log t by partial summation. Together with the estimate 9, this evaluation establishes the claim 8. Define St p t ω qpω 0 p to be the left-hand side of equation 8. Noting that ω q p 0 for all p q, we use partial summation to estimate covω q, ω 0 q<p z ω q pω 0 p p z q Sz dst + t z By equation 8, the first term satisfies Sz log log z z φq log z + O log z while z as required. q Lemma 5.4. For z >, St z log log t dt t q φq t log t + O dt t log t z log log t φq + O log log t z log log z q φq q covω 0, ω 0 log log z3 3 + O log log z. z q St t dt. + Olog log z. Proof. Now we emulate the proof of [6, Lemma.]. In preparation for a partial summation calculation, we first show that ω 0 p ωp tlog log t t log log t + O 3 log t log t p t p t 3 3

24 for all t > ; again the first equality follows from Definition 3. of ω 0. Letting l denote a variable of summation taking only prime values, and noting that at most 4 primes greater than p /5 can divide p, + O4 p t ωp p t p t l p l t /5 l p l t /5 l p l t /5 + O l t /5 l t /5 p t p mod l p mod l l t /5 l t /5 l l πt; l l, + p t l p l t /5 + O p t + p t ωp + πt t log log t πt; l, + O l t /5 log t where the error term in the last step was controlled using the q case of equation 8. Using the Bombieri Vinogradov estimate 7 in a manner similar to the argument in equation 3 now yields t log log t p t ωp l t /5 l t /5 l l lit l l + l t /5 lit l + O log t t log log t lit + O + O l l log t l t /5 l t /5 lit log log t /5 + Olog log t t log log t + O, log t which is enough to establish the claim 3. Define St p t ω 0p to be the left-hand side of equation 3, and again use partial summation to estimate covω 0, ω 0 ω 0 p z p Sz z St dst + dt. t z t q<p z By equation 3, the first term satisfies while z Sz z log log z log z log log z + O log z St z log log t log log t dt + O dt t t log t t log t z log log t3 3 + O log log t z log log z 3 3 4, + O log log z

25 as required. When we expand the hth power in the calculation of the moments M h x as in equation 36, we will need to estimate products of the additive functions ω q q 0 from Definitions.7 and 3., summed over many prime variables. Because of the presence of the multiplicative function H, defined momentarily, in these sums, it will be important how many distinct prime values are taken by these prime variables. The next three lemmas provide the details. Definition 5.5. Define a multiplicative function Hn by setting, for each prime power p γ, Hp γ p p γ + γ. p p For any prime p, we note that Hp and Hp 0; in particular, Hn 0 p p unless n is squarefull. It is easy to check that 0 Hp γ Hp for every prime p and every positive integer γ. Lemma 5.6. Let k be a positive even integer, and let 0 l k be an integer. Suppose that g g l ω 0, while the remaining functions g j l < j k equal ω qj for some prime powers q j. Then p,...,p k z p p k squarefull #{p,...,p k }k/ Hp p k g p g k p k k/! τ T k k/ covg Υ j, g Υ j + O k log log x l+k/. 33 Proof. All implicit constants in this proof may depend upon k. To each k-tuple p,..., p k counted by the sum on the left-hand side, we can uniquely associate a k/-tuple q,..., q k/ of primes satisfying q < < q k/ such that each q j equals exactly two of the p i. This correspondence defines a unique τ T k, for which τi equals the integer j such that p i q j. 5

26 Therefore, by Definition 5.5, Hp p k g p g k p k p,...,p k z p p k squarefull #{p,...,p k }k/ q k/ z q k/ / {q,...,q k/ } τ T k k/! q < <q k/ z τ T k k/! τ T k k/! τ T k Hq q k/g q τ g k q τk q,...,q k/ z q,...,q k/ distinct q,...,q k/ z q,...,q k/ distinct q,...,q k/ z q,...,q k/ distinct If we fix τ and q,..., q k/, the innermost sum over q k/ is g Υ k/q k/ g Υ k/q k/ q k/ q k/ covg Υ k/, g Υ k/ k/ covg Υ k/, g Υ k/ + O, Hq q k/g q τ g k q τk k/ g q τ g k q τk qj q j k/ g Υ jq j g Υ jq j. q j q j g Υ k/q j g Υ k/q j q j q j since each g i q log z for q z. Summing in turn over q k/,..., q in the same way, we obtain Hp p k g p g k p k p,...,p k z p p k squarefull #{p,...,p k }k/ k/! τ T k k/ covgυ j, g Υ j + O. Upon multiplying out the product corresponding to some τ T k, we obtain the leading term k/ covg Υ j, g Υ j together with terms that involve at most k/ covariances. An examination of Lemmas reveals that the order of magnitude of the leading term as a function of z is log log z l+k/, regardless of how the g j are paired with one another by τ, and that 6

27 every non-leading term is log log z l+k/ uniformly in the possibilities for the g j, We conclude that Hp p k g p g k p k p,...,p k z p p k squarefull #{p,...,p k }k/ k/! τ T k as desired where we have used z x /k in the error term. k/ covg Υ j, g Υ j + O k log log x l+k/ Lemma 5.7. Let k be a positive integer, and let g,..., g k be functions satisfying g p,..., g k p log p. Then for any i k, p,...,p k z ω 0 p i >4k log log z Hp p k g p g k p k k log z /. Proof. All implicit constants in this proof may depend upon k. Suppose that q,, q s are the distinct primes such that {p,..., p k } {q,, q s }, and let m denote any integer such that q m p i. From Definition 5.5, we know that 0 Hp p k Hq qs /q q s. Therefore, from the hypothesis on the sizes of the g j p, Hp p k g p g k p k p,...,p k z ω 0 p i >4k log log z k log z k k log z k k log z k by Mertens s theorem. Note that q m z ω 0 q m>4k log log z q m k s s m q,...,q s z ω 0 q m>4k log log z k s s m k log log z s s q m z ωq m >4k log log z q m z ω 0 q m>4k log log z s q q s q m i k i m m q m z ω 0 q m>4k log log z q m n z ωn>4k log log z q q i z i q m 34 A result of Erdős and Nicolas [5] implies that the number of n x satisfying ωn > 4k log log x is x/log x +4k log 4k 4k ; partial summation then implies that the right-hand sum is /log x 4k log 4k 4k. Equation 34 therefore implies p,...,p k z ω 0 p i >4k log log z Hp p k g p g k p k k log z k log log z k, log z 4k log 4k 4k 7 n.

28 and the lemma follows from the fact that 4k log 4k 5k > for k. Lemma 5.8. Let k be a positive integer, and let 0 l k be an integer. Suppose that g g l ω 0, while the remaining functions g j l < j k equal ω qj for some prime powers q j. When k is even, p,...,p k z p p k squarefull #{p,...,p k }<k/ while when k is odd, p,...,p k z p p k squarefull #{p,...,p k }<k/ Hp p k g p g k p k k log log x l+k/, Hp p k g p g k p k k log log x l+k /. We remark that when k is odd, the condition of summation #{p,..., p k } < k/ is always satisfied; we have nevertheless included the condition, for later convenience. Proof. All implicit constants in this proof may depend upon k. We begin by noting that by Lemma 5.7, it suffices to consider the sum on the left-hand side with the extra summation condition max ω 0 p i 4k log log z inserted. To each k-tuple p,..., p k counted by the sum on the left-hand side, we associate the positive integer s #{p,..., p k }, the primes q < < q s such that {q,..., q s } {p,..., p k }, and the integers γ,..., γ s such that q j equals exactly γ j of the p i ; note that γ γ,..., γ s is a composition, not a partition, of k, since we are not assuming any monotonicity of the γ j. Let T γ denote the set of functions from {,..., k} to {,..., s} such that for each j s, exactly γ j elements of {,..., k} are mapped to j. Given any τ T γ, define Υ j and Υ j to be two distinct preimages of j in {,..., k}; we will never need to know exactly which two preimages or to distinguish between the two. Finally, for any such τ, define l to be the number of functions among g Υ, g Υ,..., g Υ s, g Υ s that equal ω 0, and set M τ 4k log log z l l. First, observe that p,...,p k z p p k squarefull #{p,...,p k }<k/ max ω 0 p i 4k log log z Hp p k g p g k p k s<k/ q < <q s z γ,...,γ s max ω 0 q i 4k log log z γ + +γ sk Hq γ q γs s τ T γ g q τ g k q τk. By Definition 5.5, we may bound Hq γ q γs s by Hq q s. Moreover, in the innermost summand, we retain all of the factors of the form g Υ jq j and g Υ jq j while bounding all 8

29 of the other g i q τi by their pointwise upper bounds, which results in a factor of M τ : Hp p k g p g k p k p,...,p k z p p k squarefull #{p,...,p k }<k/ max ω 0 p i 4k log log z s<k/ q < <q s z γ,...,γ s γ + +γ sk s<k/ s! Hq q s τ T γ M τ s M τ q,...,q s z γ,...,γ s τ T γ γ + +γ sk Moving the sum over the q j to the inside and summing, we obtain Hp p k g p g k p k p,...,p k z p p k squarefull #{p,...,p k }<k/ max ω 0 p i 4k log log z s<k/ s! M τ γ,...,γ s τ T γ γ + +γ sk s g Υ jq j g Υ jq j g Υ jq j g Υ jq j q j q j. s covg Υ j, g Υ j. An examination of Lemmas reveals that each product on the right-hand side is log log z s+l regardless of how the g j are paired with one another by τ; consequently, s covg Υ j, g Υ j k log log z l l log log z s+l M τ { log log z s+l log log x k/ +l, log log x k /+l, if k is even, if k is odd. The lemma follows upon summing over τ, the γ i and s, which results in a constant that depends only on k. We are now ready to establish the main result of this section, which will be used repeatedly in Section 6. Recall that the functions f p and F g were defined in Definition 3.. Proposition 5.9. Let k be a positive even integer, and let 0 l k be an integer. Suppose that g g l ω 0, while the remaining functions g j l < j k equal ω qj for some prime powers q j. When k is even, k F gj n while when k is odd, x k/! τ T k k/ covg Υ j, g Υ j + O k xlog log x l+k/, k F gj n k xlog log x l+k /. 9

30 Proof. All implicit constants in this proof may depend upon k. Expanding out the left-hand side using Definition 3. results in k k g j pf p n F gj n p z f p p k n g p g k p k p,...,p k z g p g k p k Hp p k x + O ωp p k, p,...,p k z where the last equality follows from [8, equation before equation 9] with a slight change of notation. Each ω q is bounded by, while ω 0 p ωp is trivially bounded by log p/ log ; in particular, g j p j log z for all p j z. Therefore and so p,...,p k z g p g k p k ωp p k k F gj n x p,...,p k z p,...,p k z k log z πz log z k z k x, Hp p k g p g k p k + O x. Since Hp p k vanishes unless p p k is squarefull by Definition 5.5, there are at most k/ distinct primes among p,..., p k, and so we can write Hp p k g p g k p k Hp p k g p g k p k p,...,p k z p,...,p k z p p k squarefull #{p,...,p k }k/ + p,...,p k z p p k squarefull #{p,...,p k }<k/ The proposition now follows upon appealing to Lemmas 5.6 and Calculating the moments Hp p k g p g k p k. We are now ready to carry out, for h, the computation of the moments M h x. In particular, the proof of Proposition.3 requires some preparatory work, which we organize into Lemmas We also find an asymptotic formula for the function Dx in Proposition 6.5; together with Lemmas 6.4 and 6.6, this calculation reveals the origins of the perhaps mysterious constants A, B, and C appearing in Theorem.. Finally, we proof Proposition.3 at the end of this section. Recall that X log log x / log log log x, a notation that will persist throughout this section; we shall always assume that X. As our starting point, we define S ΛqF ωq n and S Λqµω q F ωq n 35 q X 30 q X