Orders of reductions of elliptic curves with many and few prime factors
aa r X i v : . [ m a t h . N T ] N ov ORDERS OF REDUCTIONS OF ELLIPTIC CURVES WITH MANYAND FEW PRIME FACTORS
LEE TROUPE
Abstract.
In this paper, we investigate extreme values of ω ( E ( F p )), where E/ Q is anelliptic curve with complex multiplication and ω is the number-of-distinct-prime-divisorsfunction. For fixed γ >
1, we prove that { p ≤ x : ω ( E ( F p )) > γ log log x } = x (log x ) γ log γ − γ + o (1) . The same result holds for the quantity { p ≤ x : ω ( E ( F p )) < γ log log x } when0 < γ <
1. The argument is worked out in detail for the curve E : y = x − x , and wediscuss how the method can be adapted for other CM elliptic curves. Introduction
Let E/ Q be an elliptic curve. For primes p of good reduction, one has E ( F p ) ≃ Z /d p Z ⊕ Z /e p Z where d p and e p are uniquely determined natural numbers such that d p divides e p . Thus, E ( F p ) = d p e p . We concern ourselves with the behavior ω ( E ( F p )), where ω ( n ) denotesthe number of distinct prime factors of the number n , as p varies over primes of goodreduction. Work has been done already in this arena: If the curve E has CM, Cojocaru[Coj05, Corollary 6] showed that the normal order of ω ( E ( F p )) is log log p , and a yearlater, Liu [Liu06] established an elliptic curve analogue of the celebrated Erd˝os - Kactheorem: For any elliptic curve E/ Q with CM, the quantity ω ( E ( F p )) − log log p √ log log p has a Gaussian normal distribution. In particular, ω ( E ( F p )) has normal order log log p and standard deviation √ log log p . (These results hold for elliptic curves without CM, ifone assumes GRH.)In light of the Erd˝os - Kac theorem, one may ask how often ω ( n ) takes on extremevalues, e.g. values greater than γ log log n , for some fixed γ >
1. A more precise versionof the following result appears in [EN79]; its proof is due to Delange.
Theorem 1.1.
Fix γ > . As x → ∞ , { n ≤ x : ω ( n ) > γ log log x } = x (log x ) γ log γ − γ + o (1) . Presently, we establish an analogous theorem for the quantity ω ( E ( F p )), where E/ Q is an elliptic curve with CM. Theorem 1.2.
Let E/ Q be an elliptic curve with CM. For γ > fixed, { p ≤ x : ω ( E ( F p )) > γ log log x } = x (log x ) γ log γ − γ + o (1) . The same result holds for the quantity { p ≤ x : ω ( E ( F p )) < γ log log x } when < γ < . The author was partially supported by NSF RTG Grant DMS-1344994.
In what follows, the above theorem will be proved for E/ Q with E : y = x − x .Essentially the same method can be used for any elliptic curve with CM; refer to thediscussion in § § §
4, respectively.
Remark.
One can ask similar questions about other arithmetic functions applied to E ( F p ). For example, Pollack has shown [Polar] that, if E has CM, then X ′ p ≤ x τ ( E ( F p )) ∼ c E · x, where the sum is restricted to primes p of good ordinary reduction for E . Several elementsof Pollack’s method of proof will appear later in this manuscript. Notation. K will denote an extension of Q with ring of integers Z K . For each ideal a ⊂ Z K , we write k a k for the norm of a (that is, k a k = Z K / a ) and Φ( a ) = Z K / a ) × .The function ω applied to an ideal a ⊂ Z K will denote the number of distinct primeideals appearing in the factorization of a into a product of prime ideals. For α ∈ Z K , k α k and Φ( α ) denote those functions evaluated at the ideal ( α ). If α is invertible modulo anideal u ⊂ Z K , we write gcd( α, u ) = 1. The notation log k x will be used to denote the k th iterate of the natural logarithm; this is not to be confused with the base- k logarithm.The letters p and q will be reserved for rational prime numbers. We make frequent useof the notation ≪ , ≫ and O -notation, which has its usual meaning. Other notation maybe defined as necessary. Acknowledgements.
The author thanks Paul Pollack for a careful reading of thismanuscript and many helpful suggestions.2.
Useful propositions
One of our primary tools will be a version of Brun’s sieve in number fields. Thefollowing theorem can be proved in much the same way that one obtains Brun’s puresieve in the rational integers, cf. [Pol09, § Theorem 2.1.
Let K be a number field with ring of integers Z K . Let A be a finitesequence of elements of Z K , and let P be a finite set of prime ideals. Define S ( A , P ) := { a ∈ A : gcd( a, P ) = 1 } , where P := Y p ∈P p . For an ideal u ⊂ Z K , write A u := { a ∈ A : a ≡ u ) } . Let X denote anapproximation to the size of A . Suppose δ is a multiplicative function taking values in [0 , , and define a function r ( u ) such that A u = Xδ ( u ) + r ( u ) for each u dividing P . Then, for every even m ∈ Z + , S ( A , P ) = X Y p ∈P (1 − δ ( p )) + O (cid:18) X u | P , ω ( u ) ≤ m | r ( u ) | (cid:19) + O (cid:18) X X u | P , ω ( u ) ≥ m δ ( u ) (cid:19) . All implied constants are absolute.
In our estimation of O -terms arising from the use of Proposition 2.1, we will makefrequent use of the following analogue of the Bombieri-Vinogradov theorem, which westate for an arbitrary imaginary quadratic field K/ Q with class number 1. For α ∈ Z K and an ideal q ⊂ Z K , write π ( x ; q , α ) = { µ ∈ Z K : k µ k ≤ x, µ ≡ α (mod q ) } . XTREME VALUES OF ω ( E ( F p )) 3 Proposition 2.2.
For every
A > , there is a B > so that X k q k≤ x / (log x ) − B max α :gcd( α, u )=1 max y ≤ x | π ( y ; q , α ) − w K · Li( y )Φ( q ) | ≪ x (log x ) A , where the above sum and maximum are taken over q ⊂ Z K and α ∈ Z K . Here w K denotesthe size of the group of units of Z K The above follows from Huxley’s analogue of the Bombieri-Vinogradov theorem fornumber fields [Hux71]; see the discussion in [Polar, Lemma 2.3].The following proposition is an analogue of Mertens’ theorem for imaginary quadraticfields. It follows immediately from Theorem 2 of [Ros99].
Proposition 2.3.
Let K/ Q be an imaginary quadratic field and let α K denote the residueof the associated Dedekind zeta function, ζ K ( s ) , at s = 1 . Then Y k p k≤ x (cid:16) − k p k (cid:17) − ∼ e γ α K log x, where the product is over all prime ideals p in Z K . Here (and only here), γ is the Euler-Mascheroni constant. Note also that the “additive version” of Mertens’ theorem, i.e., X k p k≤ x k p k = log x + B K + O K (cid:18) x (cid:19) for some constant B K , holds in this case as well; it appears as Lemma 2.4 in [Rosen].Finally, we will make use of the following estimate for elementary symmetric functions[HR83, p. 147, Lemma 13]. Lemma 2.4.
Let y , y , . . . , y M be M non-negative real numbers. For each positive integer d not exceeding M , let σ d = X ≤ k Theorem 3.1. Let E be the elliptic curve E : y = x − x and fix γ > . Then { p ≤ x : ω ( E ( F p )) > γ log x } ≪ γ x (log x ) (log x ) γ log γ − γ . The same statement is true if instead < γ < and the strict inequality is reversed onthe left-hand side. Before proving Theorem 3.1, we refer to [JU08, Table 2] for the following useful factconcerning the numbers E ( F p ): For primes p ≤ x with p ≡ E ( F p ) = p + 1 − ( π + π ) = ( π − π − , (1)where π ∈ Z [ i ] is chosen so that p = ππ and π ≡ i ) ). (Such π are sometimescalled primary .) This determines π completely up to conjugation. LEE TROUPE We begin the proof of Theorem 3.1 with the following lemma, which will allow us todisregard certain problematic primes p . Lemma 3.2. Let x ≥ and let P ( n ) denote the largest prime factor of n . Let X denotethe set of n ≤ x for which either of the following properties fail: (i) P ( n ) > x / x (ii) P ( n ) ∤ n .Then, for any A > , the size of X is O ( x/ (log x ) A ) . The following upper bound estimate of de Bruijn [dB66, Theorem 2] will be useful inproving the above lemma. Proposition 3.3. Let x ≥ y ≥ satisfy (log x ) ≤ y ≤ x . Whenever u := log x log y → ∞ , wehave Ψ( x, y ) ≤ x/u u + o ( u ) . Proof of Lemma 3.2. If n ∈ X , then either (a) P ( n ) ≤ x / x or (b) P ( n ) > x / x and P ( n ) | n . By Proposition 3.3, the number of n ≤ x for which (a) holds is O ( x/ (log x ) A ) for any A > 0, noting that (log x ) A ≪ (log x ) log x = (log x ) log x . Thenumber of n ≤ x for which (b) holds is ≪ x X p>x / x p − ≪ x exp( − log x/ x ) , and this is also O ( x/ (log x ) A ). (cid:3) We would like to use Lemma 3.2 to say that a negligible amount of the numbers E ( F p ),for p ≤ x , belong to X . The following lemma allows us to do so. Lemma 3.4. The number of p ≤ x with E ( F p ) ∈ X is O ( x/ (log x ) B ) , for any B > .Proof. Suppose E ( F p ) = b ∈ X . Then, by (1), b = k π − k , where π ∈ Z [ i ] is aGaussian prime lying above p . Thus, the number of p ≤ x with E ( F p ) = b is boundedfrom above by the number of Gaussian integers with norm b , which, by [HW00, Theorem278], is 4 P d | b χ ( d ), where χ is the nontrivial character modulo 4. Now, using the Cauchy-Schwarz inequality and Lemma 3.2,4 X b ∈X X d | b χ ( d ) ≤ X b ∈X τ ( b ) ≤ (cid:16) X b ∈X (cid:17) / (cid:16) X b ∈X τ ( b ) (cid:17) / ≪ (cid:16) x (log x ) A (cid:17) / (cid:16) x log x (cid:17) / = x (log x ) A/ − / . Since A > (cid:3) For k a nonnegative integer, define N k to be the number of primes p ≤ x of goodordinary reduction for E such that E ( F p ) possesses properties ( i ) and ( ii ) from theabove lemma and such that ω ( E ( F p )) = k . Then, in the case when γ > { p ≤ x : ω ( E ( F p )) > γ log log x } = X k>γ log x N k + O (cid:16) x (log x ) A (cid:17) for any A > 0. Our task is now to bound N k from above in terms of k . Evaluating thesum on k then produces the desired upper bound. XTREME VALUES OF ω ( E ( F p )) 5 It is clear that N k ≤ X a ≤ x − / x ω ( a )= k − X p ≤ xp ≡ a | E ( F p ) E ( F p ) /a prime . (2)To handle the inner sum, we need information on the integer divisors of E ( F p ),where p ≤ x and p ≡ a | E ( F p ) if and only if a | ( π − π − 1) = k π − k . With this inmind, we have X a ≤ x − / x ω ( a )= k − X p ≤ xp ≡ a | E ( F p ) E ( F p ) /a prime X a ≤ x − / x ω ( a )= k − X ′ π : k π k≤ xπ ≡ i ) ) a |k π − kk π − k /a prime , where the ′ on the sum indicates a restriction to primes π lying over rational primes p ≡ Divisors of shifted Gaussian primes. The conditions on the primed sum abovecan be reformulated purely in terms of Gaussian integers. Definition 3.5. For a given integer a ∈ N , write a = Q q q v q , with each q prime. For each q | a with q ≡ q = π q π q . Define a set S a which consists of all products α of the form α = (1 + i ) v Y q | aq ≡ q ⌈ v q / ⌉ Y q | aq ≡ α q , where α q ∈ { π iq π v q − iq : i = 0 , , . . . , v q } .Notice that the condition a | k π − k is equivalent to π − S a . We can therefore write X a ≤ x − / x ω ( a )= k − X p ≤ xp ≡ a | E ( F p ) E ( F p ) /a prime ≤ X a ≤ x − / x ω ( a )= k − X α ∈ S a X ′ π : k π k≤ xπ ≡ i ) ) α | π − k π − k /a prime . (3)Now, for any α ∈ S a , we have αα = a Y q ≡ q ⌈ v q / ⌉− v q . Observe that k π − k a = ( π − π − αα Y q ≡ q ⌈ v q / ⌉− v q . Therefore, if k π − k a is to be prime, the number a must satisfy exactly one of the followingproperties:1. The number a is divisible by exactly one prime q ≡ v q an oddnumber, and α = u ( π − 1) where u ∈ Z [ i ] is a unit; or2. All primes q ≡ a have v q even, and ( π − /α is a primein Z [ i ]. LEE TROUPE This splits the outer sum in (3) into two components. Lemma 3.6. We have X ♭a ≤ x − / x ω ( a )= k − X α ∈ S a X ′ π : k π k≤ xπ ≡ i ) )( π − /α ∈ U O (cid:18) x log A x (cid:19) , where U is the set of units in Z [ i ] and the ♭ on the outer sum indicates a restriction tointegers a such that there is a unique prime power q v q k a with q ≡ and v q odd.Proof. If α = u ( π − 1) for u ∈ U , then there are at most four choices for π , given α . Thus X ♭a ≤ x − / x ω ( a )= k − X α ∈ S a X ′ π : k π k≤ xπ ≡ i ) ) α = u ( π − ≤ X ♭a ≤ x − / x ω ( a )= k − | S a | . We have | S a | = Q q ≡ ( v q + 1); this is bounded from above by the divisor functionon a , which we denote τ ( a ). Therefore, the above is ≪ X a ≤ x − / x τ ( a ) ≪ x − / x (log x ) , which is O ( x/ log A x ) for any A > (cid:3) The second case provides the main contribution to the sum. Lemma 3.7. Let a ≤ x − / x with ω ( a ) = k − such that all primes q ≡ dividing a have v q even. Let α ∈ S a . Then X ′ π : k π k≤ xπ ≡ i ) ) α | π − π − /α prime ≪ x (log x ) k α k (log x ) uniformly over all a as above and α ∈ S a .Proof. If π ≡ α ), then π = 1 + αβ for some β ⊂ Z [ i ]. Thus β = π − α , and so k β k ≤ x k α k . Let A denote the sequence of elements in Z [ i ] given by n β (1 + αβ ) : k β k ≤ x k α k o . Define P = { p ⊂ Z [ i ] : k p k ≤ z } where z is a parameter to be chosen later. Then, in thenotation of Theorem 2.1, X ′ π : k π k≤ xπ ≡ i ) ) α | π − π − /α prime ≤ S ( A , P ) + O ( z ) . Here, the O ( z ) term comes from those π ∈ Z [ i ] such that both π and ( π − /α are primesof norm less than z .For u ⊂ Z [ i ], write A u = { a ∈ A : a ≡ u ) } . An element a ∈ A is counted by A u if and only if a generator of u divides a . Thus, by familiar estimates on the numberof integer lattice points contained in a circle, A u satisfies the equation A u = 2 πx k α k ν ( u ) k u k + O (cid:16) ν ( u ) √ x ( k α kk u k ) / (cid:17) , XTREME VALUES OF ω ( E ( F p )) 7 where ν ( u ) = { β (mod u ) : β (1 + αβ ) ≡ u ) } . We apply Theorem 2.1 with X = 2 πx k α k and δ ( u ) = ν ( u ) k u k . With these choices, we have r ( u ) = O (cid:16) ν ( u ) √ x ( k α kk u k ) / | (cid:17) . Then, for any even integer m ≥ S ( A , P ) = 2 πx k α k Y k p k≤ z (cid:18) − ν ( p ) k p k (cid:19) + O (cid:18) √ x k α k / X u | P ω ( u ) ≤ m ν ( u ) k u k / (cid:19) (4) + O (cid:18) x k α k X u | P ω ( u ) ≥ m δ ( u ) (cid:19) , where P = Q p ∈P p .For a prime p , we have ν ( p ) = 2 if α p ) and ν ( p ) = 1 otherwise. Therefore,the product in the first term is Y k p k≤ z p ∤ ( α ) (cid:18) − k p k (cid:19) Y k p k≤ z p | ( α ) (cid:18) − k p k (cid:19) ≤ Y k p k≤ z (cid:18) − k p k (cid:19) Y k p k≤ z p | ( α ) (cid:18) − k p k (cid:19) − ≪ z ) k α k Φ( α ) , where in the last step we used Proposition 2.3.Choose z = x x )2 . Then our first term in (4) is ≪ x (log x ) Φ( α )(log x ) . Recall that k α k = a , and a ≤ x − / x . Since Φ( α ) ≫ k α k / log x (analogous to theminimal order for the usual Euler function, c.f. [HW00, Theorem 328]), the above is ≪ x (log x ) k α k (log x ) . We now show that this “main” term dominates the two O -terms uniformly for α ∈ S a and a ≤ x − / x . For the first O -term, we begin by noting that ν ( u ) / k u k / ≪ m = 10 ⌊ log x ⌋ , we have X u | P ω ( u ) ≤ m ν ( u ) k u k / ≪ m X k =0 (cid:18) π K ( z ) k (cid:19) ≤ m X k =0 π K ( z ) k ≤ π K ( z ) m ≤ x / 20 log x , where π K ( z ) denotes the number of prime ideals p ⊂ Z [ i ] with norm up to z . Therefore,the inequality x (log x ) k α k (log x ) ≫ x / / 20 log x k α k / LEE TROUPE holds for all α with k α k ≤ x − / x , as desired.Next we handle the second O -term. The sum in this term is X u | P ω ( u ) ≥ m δ ( u ) ≤ X s ≥ m s ! (cid:16) X k p k≤ z ν ( p ) k p k (cid:17) s . Observe that, by Proposition 2.3, we have X k p k≤ z ν ( p ) k p k ≤ x + O (1) . Thus, by the ratio test, one sees that the sum on s is ≪ m ! (2 log x + O (1)) m . Using Proposition 2.3 followed by Stirling’s formula, we obtain that the above quantityis 1 m ! (2 log x + O (1)) m ≤ (cid:16) e log x + O (1)10 ⌊ log x ⌋ (cid:17) ⌊ log x ⌋ ≪ (cid:16) e (cid:17) x ≤ x ) . So the second O -term is ≪ x k α k (log x ) , and this is certainly dominated by the main term. (cid:3) From Lemmas 3.6 and 3.7, we see (2) can be rewritten N k ≪ x (log x ) (log x ) X a ≤ x − / x ω ( a )= k − | S a | a + O (cid:16) x log A x (cid:17) , noting that k α k = a for all a under consideration and all α ∈ S a . We are now in aposition to bound N k from above in terms of k . Lemma 3.8. We have X a ≤ x − / x ω ( a )= k − | S a | a ≤ (log x + O (1)) k − ( k − . Proof. We have already seen that the size of S a is Q p | a : p ≡ ( v p + 1), where v p isdefined by p v p k a . Recall that in the current case, each prime p ≡ a appears to an even power. Therefore, we have X a ≤ xω ( a )= k − | S a | a ≤ k − X p ℓ ≤ xp | S p ℓ | p ℓ + X p k ≤ xp ≡ | S p k | p k + O (1) ! k − . (5)Note that | S p k | = 1 for each prime p ≡ O (1) term, giving X a ≤ xω ( a )= k − | S a | a ≪ k − X p ℓ ≤ xp | S p ℓ | p ℓ + O (1) ! k − . (6) XTREME VALUES OF ω ( E ( F p )) 9 Now X p ℓ ≤ xp | S p ℓ | p ℓ = X p ℓ ≤ xp ≡ ℓ + 1 p ℓ + O (1)= X p ≤ xp ≡ p + O (1)= log x + O (1) . Inserting this expression into (6) proves the lemma. (cid:3) Finishing the upper bound. We have shown so far that N k ≪ x (log x ) (log x ) · (log x + O (1)) k − ( k − . We now sum on k > γ log x for fixed γ > < γ < X k>γ log x (log x + O (1)) k − ( k − ≪ (cid:18) e log x + O (1) ⌊ γ log x ⌋ (cid:19) ⌊ γ log x ⌋ ≪ (cid:18) eγ (cid:16) O (cid:16) x (cid:17)(cid:17)(cid:19) ⌊ γ log x ⌋ ≪ (cid:16) eγ (cid:17) ⌊ γ log x ⌋ ≪ γ (log x ) γ − γ log γ . Thus, we have obtained an upper bound of ≪ γ x (log x ) (log x ) γ log γ − γ , as desired. 4. A lower bound Theorem 4.1. Consider E : y = x − x and fix γ > . Then { p ≤ x : ω ( E ( F p )) > γ log x } ≥ x (log x ) γ log γ − γ + o (1) . The same statement is true if instead < γ < and the strict inequality is reversed onthe left-hand side. Our strategy in the case γ > E ( F p ) = k π − k ,where π ≡ i ) ) and p = ππ . Let k be an integer to be specified later and fixan ideal s ∈ Z [ i ] with the following properties:(A) ((1 + i ) ) | s (B) ω ( s ) = k (C) P + ( k s k ) ≤ x / γ log x (D) Each prime ideal p | s (with the exception of (1 + i )) lies above a rational prime p ≡ p dividing s lie above distinct p (F) s squarefreeHere P + ( n ) denotes the largest prime factor of n . Note that we have ω ( s ) = ω ( k s k ).First, we will estimate from below the size of the set M s , defined to be the set of those π ∈ Z [ i ] with k π k ≤ x satisfying the following properties: (1) π prime (in Z [ i ])(2) k π k prime (in Z )(3) π ≡ s )(4) P − (cid:16) k π − kk s k (cid:17) > x / γ log x .Here P − ( n ) denotes the smallest prime factor of n . The conditions on the size of theprime factors of k s k and k π − k / k s k imply that each π with k π k ≤ x belongs to at mostone of the sets M s . If k is chosen to be greater than γ log x , then carefully summingover s satisfying the conditions above yields a lower bound on the count of distinct π corresponding to p with the property that ω ( E ( F p )) ≥ k > γ log x . The problem ofcounting elements π and π with p = ππ is remedied by inserting a factor of , which isof no concern for us.More care is required in the case 0 < γ < 1, which is handled in Section 4.3.4.1. Preparing for the proof of Theorem 4.1. Suppose the fixed ideal s is generatedby σ ∈ Z [ i ]. We will estimate from below the size of M s using Theorem 2.1. Define A to be the sequence of elements of Z [ i ] of the form n π − σ : k π k ≤ x, π prime, and π ≡ σ ) o . Let P denote the set of prime ideals { p : k p k ≤ z } , where z := x / γ log x . Let P := Q p ∈P p . If π − σ ≡ p ) implies k p k ≥ z , then all primes p | k π − σ k have p >x / γ log x . Note also that if a prime π ∈ Z [ i ], k π k ≤ x is such that k π k is not prime,then k π k = p for some rational prime p , and so the count of such π is clearly O ( √ x ).Therefore, we have M s ≥ S ( A , P ) + O ( √ x ) . Lemma 4.2. With M s defined as above, we have M s ≥ c · Li( x ) log x Φ( s ) log x + O (cid:18) X u | P ω ( u ) ≤ m | r ( us ) | (cid:19) + O (cid:18) s ) Li( x )(log x ) (cid:19) + O ( √ x ) , where r ( v ) = | Li( x )Φ( v ) − π ( x ; v , | and c > is a constant.Proof. First, note that we expect the size of A to be approximately X := 4 Li( x )Φ( s ) . Write A u = { a ∈ A : u | a } . Then A u = Xδ ( u ) + r ( us ) , where δ ( u ) = Φ( s )Φ( us ) and r ( us ) = | Li( x )Φ( us ) − π ( x ; us , | . By Theorem 2.1, for any even integer m ≥ S ( A , P ) = 4 Li( x )Φ( s ) Y k p k≤ z (cid:18) − Φ( s )Φ( ps ) (cid:19) + O (cid:18) X u | P ω ( u ) ≤ m | r ( us ) | (cid:19) + O (cid:18) Li( x )Φ( s ) X u | P ω ( u ) ≥ m δ ( u ) (cid:19) . XTREME VALUES OF ω ( E ( F p )) 11 Using Proposition 2.3, we have Y k p k≤ z (cid:18) − Φ( s )Φ( ps ) (cid:19) = Y k p k≤ z p ∤ s (cid:18) − p ) (cid:19) Y k p k≤ z p | s (cid:18) − k p k (cid:19) = Y k p k≤ z (cid:18) − k p k (cid:19) Y k p k≤ z p ∤ s (cid:18) − k p k − (cid:19) ≫ z = log x log x . Take m = 14 ⌊ log x ⌋ . We leave aside the first O -term and concentrate for now on thesecond. This term is handled in essentially the same way as in the proof of the upperbound: The sum in the this term is bounded from above by X s ≥ m s ! (cid:16) X k p k≤ z δ ( p ) (cid:17) s . By Proposition 2.3, we have X k p k≤ z δ ( p ) ≤ log x + O (1) . Now, one sees once again by the ratio test that the sum on s is ≪ m ! (cid:16) X k p k≤ z δ ( p ) (cid:17) m ≤ m ! (log x + O (1)) m . Thus, by the same calculations as in the proof of Theorem 3.1, the second O -term is ≪ Li( x )Φ( s )(log x ) , completing the proof of the lemma. (cid:3) We now sum this estimate over σ in an appropriate range to deal with the O -termsand establish a lower bound. Here, the cases γ > < γ < The case γ > . The argument in this case is somewhat simpler. Recall that s is chosen to satisfy properties A through F listed below Theorem 4.1; in particular, ω ( s ) = k for some integer k and P + ( k s k ) ≤ x / γ log x . Choose k := ⌊ γ log x ⌋ + 2.Since ω ( k s k ) = ω ( s ), we have that k s k ≤ x k/ γ log x ≤ x / . A lower bound follows byestimating the quantity M = X ′ s M s , where the prime indicates a restriction to those ideals s ⊂ Z [ i ] satisfying properties Athrough F mentioned above. Lemma 4.3. We have M ≫ x log x (log x + O (log x )) k k !(log x ) . Proof. Since P k s k≤ x / Φ( s ) ≪ log x , the second O -term in Lemma 4.2 is, upon summingon s , bounded by a constant times Li( x ) / (log x ) . The third error term, O ( √ x ), istherefore safely absorbed by this term. We now handle the sum over s of the first O -term. We have | r ( us ) | = | π ( x ; us , − Li( x )Φ( us ) | . We can think of the double sum (over s and u ) as a single sum over a modulus q ,inserting a factor of τ ( q ) to account for the number of ways of writing q as a product oftwo ideals in Z [ i ]. (Here, τ ( q ) is the number of ideals in Z [ i ] which divide q .) Recallingour choice of m = 14 ⌊ log x ⌋ , we have X k s k≤ x / X u | P ω ( u ) ≤ m | r ( us ) | ≪ X k q k 1. Now, for all y > i ⊂ Z [ i ] we have π ( y ; i , ≪ y/ k i k ; indeed, the same inequality is true with π ( y ; i , 1) replaced by the count of all properideals ≡ i ). Thus (cid:12)(cid:12)(cid:12) π ( x ; q , − x )Φ( q ) (cid:12)(cid:12)(cid:12) ≪ x Φ( q ) . Using this together with the Cauchy-Schwarz inequality and Proposition 2.2, we see that,for any A > X k q k 12 log x − x + O (1) ! k − × − (cid:18) k − (cid:19)(cid:16) S (cid:17) X p ∈P p ) ! . The quantity (cid:0) k − (cid:1) is bounded from above by ⌈ γ log x ⌉ , and the sum on 1 / Φ( p ) tendsto 0 as x → ∞ . Therefore,1 − (cid:18) k − (cid:19)(cid:16) S (cid:17) X p ∈P p ) ≥ − γ X p ∈P p ) ≥ x , and so x log x (log x ) X ′ s s ) ≫ x log x (log x + O (log x )) k − ( k − x ) , as desired. (cid:3) With k = ⌊ γ log x ⌋ + 2 and by the more precise version of Stirling’s formula n ! ∼√ πn ( n/e ) n , we have(log x + O (log x )) k − ( k − ≫ p log x (cid:18) e log x + O (log x ) ⌊ γ log x ⌋ (cid:19) ⌈ γ log x ⌉ = 1 p log x (cid:18) eγ (cid:16) O (cid:16) log x log x (cid:17)(cid:17)(cid:19) ⌈ γ log x ⌉ = (log x ) γ − γ log γ + o (1) . This yields a main term of the shape x (log x ) γ log γ − γ + o (1) , which completes the proof of Theorem 4.1 in the case γ > The case 0 < γ < Above, we used the fact that if π − s ⊂ Z [ i ] with ω ( k s k ) = k , then k π − k will have at least k > γ log x prime factors. Thecase 0 < γ < k π − k / k s k does not have too many prime factors. Lemma 4.4. For any s ⊂ Z [ i ] satisfying properties A through F listed below Theorem4.1, we have { π ∈ M s : ω (cid:18) k π − kk s k (cid:19) > log x log x } ≪ x k s k (log x ) A . Upon discarding those π counted by the above lemma, the remaining π will have theproperty that ω ( k π − k ) ∈ [ k, k + log x/ log x ]. Choosing k to be the greatest integerstrictly less than γ log x − log x/ log x ensures that k π − k < γ log x . Proof of Lemma 4.4. We begin with the observation that, for any s ⊂ Z [ i ] under consid-eration and π ∈ M s , we have k π − k / k s k ≤ x/ k s k . Therefore, we estimate X k a k≤ x k s k ω ( k a k ) > log x/ log xP − ( k a k ) >x / γ log2 x ≤ x k s k X k a k≤ x k s k ω ( k a k ) > log x/ log xP − ( k a k ) >x / γ log2 x k a k . Noting that ω ( k a k ) ≤ ω ( a ) for any a ⊂ Z [ i ], by Theorem 2.3 and Stirling’s formula, wehave X k a k≤ x k s k ω ( k a k ) > log x/ log xP − ( k a k ) >x / 100 log2 x k a k ≤ X k a k≤ x k s k ω ( a ) > log x/ log xP − ( k a k ) >x / 100 log2 x k a k≤ X ℓ> log x/ log x ℓ ! (cid:16) X x / 100 log2 x ≤k p k≤ x k s k ∞ X m =1 k p k m (cid:17) ℓ ≪ X ℓ> log x/ log x (cid:16) e log x + O (1) ℓ (cid:17) ℓ . For each ℓ > log x/ log x , we have ( e log x + O (1)) /ℓ < / 2. Thus X ℓ> log x/ log x (cid:16) e log x + O (1) ℓ (cid:17) ℓ ≪ (cid:16) e log x + O (1) ⌊ log x/ log x ⌋ + 1 (cid:17) ⌊ log x/ log x ⌋ +1 ≪ (cid:16) x ) o (1) (cid:17) log x/ log x ≪ e − x log x/ log x . This last expression is smaller than (log x ) − A , for any A > 0. Therefore, for any fixed A > 0, { π ∈ M s : ω (cid:18) k π − kk s k (cid:19) > log x log x } ≪ x k s k (log x ) A . (cid:3) Write M ′ s = { π ∈ M s : ω (cid:18) k π − kk s k (cid:19) ≤ log x log x } . Lemmas 4.2 and 4.4 show that M ′ s satisfies M ′ s ≥ c · x log x Φ( s )(log x ) + O (cid:18) X u | P ω ( u ) ≤ m | r ( us ) | (cid:19) + O (cid:18) s ) Li( x )(log x ) (cid:19) + O (cid:18) x k s k (log x ) A (cid:19) + O ( √ x ) , for any A > 0. Here, all quantities are defined as in the previous section. Just asbefore, we sum this quantity over s ⊂ Z [ i ] satisfying conditions A through F listed below XTREME VALUES OF ω ( E ( F p )) 15 Theorem 4.1. Letting ′ on a sum indicate a restriction to such s , we have, by the samecalculations as before, M ′ ≫ x log x (log x + O (log x )) k − ( k − x ) , where M ′ = X ′ s M ′ s . Recall that k is chosen to be the largest integer strictly less than γ log x − log x/ log x ;then by Stirling’s formula,(log x + O (log x )) k − ( k − ≫ p log x (cid:16) e log x + O (log x ) k − (cid:17) k − ≫ p log x (cid:16) eγ (cid:16) O (cid:0) x (cid:1)(cid:17) γ log x − log x/ log x − ≫ (log x ) γ log γ − γ + o (1) . 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