On the size of primitive sets in function fields
Andrés Gómez-Colunga, Charlotte Kavaler, Nathan McNew, Mirilla Zhu
aa r X i v : . [ m a t h . N T ] J a n ON THE SIZE OF PRIMITIVE SETS IN FUNCTION FIELDS
ANDR´ES G ´OMEZ-COLUNGA, CHARLOTTE KAVALER, NATHAN MCNEW, AND MIRILLA ZHU
Abstract.
A set is primitive if no element of the set divides another. We consider primitive sets ofmonic polynomials over a finite field and find natural generalizations of many of the results knownfor primitive sets of integers. In particular, we show that primitive sets in the function field havelower density zero by showing that the sum P a ∈ A q deg a deg a , an analogue of a sum considered byErd˝os, is uniformly bounded over all primitive sets A . We then adapt a method of Besicovitch toconstruct primitive sets in F q [ x ] with upper density arbitrarily close to q − q and generalize a resultof Martin and Pomerance on the asymptotic growth rate of the counting function of a primitiveset. Along the way we prove a quantitative analogue of the Hardy-Ramanujan theorem for functionfields, as well as bounds on the size of the k -th irreducible polynomial. Introduction
We call a set primitive if no element of the set divides another. A natural question to ask ofprimitive sets is how large they can be. In the integers, this has been answered for various notionsof size. Erd˝os [8] proved in 1935 that primitive sets have lower density zero by showing that X n ∈ A n log n (1)converges for any primitive set A = { } . In fact, Erd˝os showed that (1) is uniformly bounded andlater conjectured that it is maximized when the set A is taken to be the prime numbers. Thatsame year, Besicovitch [5] showed that there exist primitive sets A with upper density greater than − ǫ for any ǫ >
0. Because there cannot exist a primitive set having upper density greater thanor equal to , this is essentially the best possible upper density of a primitive set.In contrast to the occasionally large but usually very sparse sets in Besicovitch’s construction,Ahlswede, Khachatrian, and S´ark¨ozy [2] consider primitive sets that are large and consistentlygrowing. They show that there exists a primitive set A whose counting function A ( x ) = | [1 , x ] ∩ A | satisfies A ( x ) ≫ x log log x (log log log x ) ǫ for any ǫ >
0. This is nearly best possible, as one can show that A ( x ) = o (cid:16) x log log x (log log log x ) (cid:17) forany primitive set.Nevertheless, Martin and Pomerance make a small improvement regarding the ǫ in this result[12]. They prove that given any positive increasing function L ( x ) satisfying L ( x ) ∼ L (2 x ) suchthat R ∞ dtt log tL ( t ) < ∞ , there exists a primitive set A with counting function that satisfies A ( x ) ≍ x log log x · log log log x · L (log log x )for sufficiently large x . In particular one can take L ( x ) = log x log x · · · (log j − x ) ǫ , (wherelog k x denotes the k -fold iterated logarithm) for any j > A ( x ) ≍ x log x · log x · · · log j x · (log j +1 x ) ǫ . In this paper, we consider primitive subsets of polynomials over a finite field F q [ x ] and obtainresults analogous to those of Erd˝os, Besicovitch, and Martin and Pomerance in this setting. InCorollary 2.4, we show that the lower density of a primitive set in the function field is always zero y considering a sum analogous to (1). We then give a construction in Theorem 3.1 of a set withupper density arbitrarily close to q − q . Finally, we prove a function field analogue of Martin andPomerance’s result in Theorem 6.1, demonstrating the existence of primitive sets S ⊂ F q [ x ] withconsistently growing counting functions S ′ ( n ) = |{ f ∈ S : deg( f ) = n }| of size S ′ ( n ) ≍ q n log n · log log n · L (log n )where L ( n ) satisfies the same restrictions as in the integer case.In order to prove this result, we establish two additional results which may be of independentinterest. For any ordering { P k } of the monic irreducible polynomials over F q such that deg P k ≤ deg P j if k < j we show in Theorem 4.1 thatlog q k + log q (log q k ) + log q ( q − − o (1) ≤ deg P k ≤ log q k + log q (log q k ) + log q ( q −
1) + o (1)as k → ∞ . Then, in Proposition 5.5, we obtain bounds on the number of polynomials of degree n having either unusually few or unusually many irreducible factors. In particular, for fixed 0 < α < < β we determine that the number of degree n polynomials having at most α log n or at least β log n factors is O α (cid:16) q n n Q ( α ) √ log n (cid:17) and O β (cid:16) q n n Q ( β ) √ log n (cid:17) respectively, where Q ( y ) = y log y − y + 1.1.1. Primitive sets of polynomials.
Let M q denote the set of monic polynomials in F q [ x ] and I q denote the set of irreducible polynomials in M q . Just as in the integers, we say that a set A ⊂ M q is primitive if no element divides another. It is not difficult to see that I q is primitive; some otherexamples of primitive sets include the set of monic polynomials with exactly k irreducible factorscounted with multiplicity and the set of monic polynomials of degree n . In this paper, we willcompare the growth of primitive sets through two measures of size: counting functions and naturaldensities. Definition 1.1.
For S ⊂ F q [ x ], the counting functions of S are given by S ( n ) = { f ∈ S : deg f ≤ n } and S ′ ( n ) = { f ∈ S : deg f = n } . Definition 1.2.
The natural density of S is given by d ( S ) = lim n →∞ S ( n ) M q ( n ) , and the upper density and lower density of S are given by d ( S ) = lim sup n →∞ S ( n ) M q ( n ) and d ( S ) = lim inf n →∞ S ( n ) M q ( n ) . Lower Density of Primitive Sets
Intuitively, a primitive set cannot be too large because including any one element in the setmeans that all multiples of that element must be excluded. Here, we formalize this notion byshowing that the lower density of any primitive set A ⊂ M q must be zero. In order to do so, wegeneralize a 1935 proof of Erd˝os [8] to the function field setting. Since the result is immediate for A = { } , we will assume A = { } for the remainder of this section.Following Erd˝os, [8] our proof depends on the convergence of the sum X a ∈ A k a k deg a , (2)for all primitive A ⊂ M q , where k a k := q deg a denotes the norm of a . Because this sum is a functionfield analogue of the sum (1) Erd˝os considered in his 1935 paper, we will call this sum the Erd˝ossum of A .For any polynomial f ∈ F q [ x ], let d ( f ) denote the smallest degree of an irreducible factor of f ,and let D ( f ) denote the largest degree of an irreducible factor of f . By the Sieve of Eratosthenes, he density of the set n g ∈ M q : f | g, d (cid:16) gf (cid:17) > D ( f ) o of polynomials which are multiples of f butare not divisible by any other polynomials of degree less than D ( f ) is1 k f k Y p ∈I q deg p ≤ D ( f ) (cid:18) − k p k (cid:19) . (3)This can be derived from the fact that for any fixed polynomial p , the density of multiples of p isexactly k p k . We start by showing that the sum of densities of the form (3) ranging over all elements f contained in a primitive set A is no greater than 1. Proposition 2.1. If A is a primitive set, then X a ∈ A k a k Y p ∈I q deg p ≤ D ( a ) (cid:18) − k p k (cid:19) ≤ . Proof.
Suppose for contradiction that the inequality is false. Then there exists some N ∈ N such that X a ∈ A deg a ≤ N k a k Y p ∈I q deg p ≤ D ( a ) (cid:18) − k p k (cid:19) > . For any n ∈ N and a ∈ A , we define a n to be the number of monic polynomials of degree n divisible by a but by no b ∈ A such that D ( b ) ≤ D ( a ). Note that included in the count a n are alldegree n polynomials of the form ga with d ( g ) > D ( a ).There are q n − deg a polynomials of degree n − deg a , and using the Sieve of Erastosthenes, we seethat the number of such polynomials g is approximately q n − deg a Y p ∈I q deg p ≤ D ( a ) (cid:18) − k p k (cid:19) = q n − deg a Y p ∈I q deg p ≤ D ( a ) (cid:18) − q deg p (cid:19) , since each term in the product represents the proportion of polynomials not divisible by an irre-ducible polynomial p . If we choose n large enough so that n ≥ deg a + X p ∈I q deg p ≤ D ( a ) deg p, then the error in this approximation vanishes because all terms in the product expansion becomeintegers. Hence for sufficiently large n , a n = q n − deg a Y p ∈I q deg p ≤ D ( a ) (cid:18) − k p k (cid:19) . For any two polynomials a ′ , a ′′ ∈ A , the polynomials counted by a ′ n and a ′′ n form disjoint sets:the least irreducible factor of each polynomial counted by a ′ n and a ′′ n has degree deg a ′ and deg a ′′ ,respectively, and if deg a ′ = deg a ′′ , then each polynomial counted by a ′ n will be divisible by a ′ ,while no polynomial counted by a ′′ n will be divisible by a ′ . Hence, we can sum over elements of A to obtain q n ≥ X a ∈ A deg a ≤ N a n ≥ X a ∈ A deg a ≤ N q n − deg a Y p ∈I q deg p ≤ D ( a ) (cid:18) − k p k (cid:19) . ividing by q n gives 1 ≥ X a ∈ A deg a ≤ N k a k Y p ∈I q deg p ≤ D ( a ) (cid:18) − k p k (cid:19) , which is a contradiction since we assumed that the sum on the right hand side was strictly greaterthan 1. Thus our original assumption must have been false, and so X a ∈ A k a k Y p ∈I q deg p ≤ D ( a ) (cid:18) − k p k (cid:19) ≤ . (cid:3) An analogue of Mertens’ third theorem in function fields gives an asymptotic expression for theproduct in this expression.
Theorem 2.2. Y p ∈I q deg p ≤ n (cid:18) − k p k (cid:19) ∼ e γ n , where γ is the Euler-Mascheroni constant.Proof. The result is a special case of Theorem 3 in [14]. (cid:3)
Using this result, we can show that the Erd˝os sum (2) of any primitive set A converges, and infact is uniformly bounded. Theorem 2.3.
There exists a constant C such that X a ∈ A k a k deg a ≤ C for all primitive sets A ⊂ M q .Proof. By Theorem 2.2, there exists a constant c such that Y p ∈I q deg p ≤ n (cid:18) − k p k (cid:19) ≥ ce γ n for all positive integers n . Hence for any a ∈ A ,1 k a k Y p ∈I q deg p ≤ D ( a ) (cid:18) − k p k (cid:19) ≥ ce γ k a k D ( a ) ≥ ce γ k a k deg a . Summing over all a ∈ A gives us1 ≥ X a ∈ A k a k Y p ∈I q deg p ≤ D ( a ) (cid:18) − k p k (cid:19) ≥ ce γ X a ∈ A k a k deg a , upon which we see that X a ∈ A k a k deg a ≤ e γ c . (cid:3) Just as in the integer case, one can treat the Erd˝os sum as a measure of the size of a primitiveset which gives larger weight to polynomials of lower degree; we explore this idea further in a futurepaper. For now, we use this result to find the lower density of any primitive set.
Corollary 2.4. If A ⊂ M q is any primitive set then d ( A ) = 0 . roof. Suppose for contradiction that d ( A ) = 0. Then there exists a positive constant C so that A ( n ) ≥ C ( q n +1 − q − ≥ Cq n for all sufficiently large n . Using this and partial summation we find that X a ∈ A deg a ≤ n k a k deg a ≫ log n. This contradicts Theorem 2.3, so we can conclude that d ( A ) = 0. (cid:3) It follows immediately that if the natural density of a primitive subset of M q exists, then it mustbe equal to zero. The upper densities of such a primitive sets can be much greater however, as wesee in the next section.3. Primitive Sets with Optimal Upper Density
Primitive sets of integers cannot have upper density larger than ; this can be seen by partitioningthe integers into disjoint sets, each of which contains an odd number and all of its even multiples.Since a primitive set can include at most one element from each of these sets, its upper density canbe at most .A similar argument can be used in the function field. For some irreducible polynomial f in M q of degree 1, we partition M q into disjoint subsets of the form { f k g : k ∈ N } , where g is monic andnot divisible by f . Like in the integer case, any primitive set A can include at most one elementfrom each of these sets, which gives a maximum upper density of q − q . Here, we demonstrate theexistence of primitive sets with density arbitrarily close to this bound by generalizing Besicovitch’sconstruction from [5]. Theorem 3.1.
For any ǫ > , there exists a primitive set A ⊂ F q [ x ] with d ( A ) > q − q − ε. Proof.
We construct a primitive set A from an increasing sequence of positive integers { n i } , to bedetermined shortly, as follows. We include in A all the monic polynomials of degree n , and notethat this set contains M ′ q ( n ) M q ( n ) = M ′ q ( n ) q n + . . . + q + 1 = q n ( q n +1 − / ( q − > q − q of the monic polynomials of degree up to n . We then include all the monic polynomials of degree n , removing any polynomials having a divisor of degree n so that our set remains primitive, andrepeat this process for all n i to construct an infinite primitive set. Letting I n denote the monicpolynomials of degree n and T n denote the set of non-unit multiples of polynomials in I n , we seethat our primitive set can be written as A = ∪ ∞ i =1 ( I n i \ ∪ i − j =1 T n j ).We now define our sequence { n i } to ensure that the proportion of polynomials thrown out at eachstep is sufficiently small. For a given ε >
0, we require { n i } to satisfy two conditions: d ( T n i ) ≤ ε i +1 ,and T ni − ( n ) M q ( n ) ≤ ε i for all n ≥ n i .We construct this sequence inductively. Car shows [6] that lim n →∞ d ( T n ) = 0, so there existsan integer n such that d ( T n ) ≤ ε . Now suppose we have already found n , n , ..., n j − satisfyingthe conditions of the sequence. Then since d ( T n j − ) ≤ ε j , there exists N j such that for all n ≥ N j T n j − ( n ) M q ( n ) ≤ ε j − y the definition of upper density. Furthermore, there exists N ′ j such that d ( T n ) ≤ ε j +1 for all n ≥ N ′ j . Then if we let n j = max( n j − + 1 , N j , N ′ j ), we see that n j satisfies all the conditions ofthe sequence.We now show that for each n i , the proportion of monic polynomials of degree up to n i which arein A is at least q − q − ε : A ( n i ) M q ( n i ) ≥ | I n i \ ∪ i − j =1 T n j ) |M q ( n i ) = | I n i | − | ∪ i − j =1 ( T n j ∩ I n i ) |M q ( n i ) ≥ M ′ q ( n i ) M q ( n i ) − i − X j =1 T n j ( n i ) M q ( n i ) . We know M ′ q ( n i ) M q ( n i ) > q − q , and furthermore, i − X j =1 T n j ( n i ) M q ( n i ) ≤ i − X j =1 ε i < ε, which implies A ( n i ) M q ( n i ) ≥ q − q − ε. Because this is true for all n i , we have d ( A ) = lim sup n →∞ A ( n ) M q ( n ) ≥ q − q − ε. (cid:3) The size of the k -th irreducible polynomial Having investigated the natural densities of primitive sets, we now consider more carefully theircounting functions. In particular, we construct primitive sets with consistently large countingfunctions, in contrast to the erratically growing counting functions of our modified Besicovitchconstruction.The asymptotic growth rate of the primitive sets we construct will be closely related to thedistribution of irreducible polynomials in F q [ x ]. Define π ′ q ( n ) to be the number of irreducible monicpolynomials in M q of degree exactly n . From an exact formula for π ′ q ( n ) that Gauss derived in1797, it follows that π ′ q ( n ) ∼ q n n , (4)which can be regarded as a function field analogue of the Prime Number Theorem. In fact it followseasily from Gauss’ formula that we always have the bound π ′ q ( n ) ≤ q n n which will be used frequently below.Here, we investigate the size of the k -th irreducible polynomial if we impose a total ordering onthe irreducible polynomials, analogous to that which exists for the primes. Because the irreduciblesare already partially ordered by degree, we will require that our ordering respects degree. We let P k denote the k -th irreducible polynomial in M q under any such ordering and find bounds for boththe degree and norm of P k . Theorem 4.1. If { P k } is an arbitrary ordering of irreducible polynomials of increasing degree in M q , then as k → ∞ we can bound the degree of P k by log q k + log q (log q k ) + log q ( q − − o (1) ≤ deg P k ≤ log q k + log q (log q k ) + log q ( q −
1) + o (1) where log q denotes the logarithm base q . roof. We know that π q (deg P k − < k ≤ π q (deg P k ) , where π q ( n ) is the counting function for monic irreducible polynomials of degree up to n . Thefunction π q ( n ) is more difficult to work with than π ′ q ( n ), but Kruse and Stichtenoth [10] haveobtained an asymptotic expression for this quantity: π q ( n ) ∼ q n +1 ( q − n . Thus as k → ∞ , q deg P k ( q − P k −
1) (1 + o (1)) ≤ k ≤ q deg P k +1 ( q −
1) deg P k (1 + o (1)) . (5)Taking the base- q logarithm of both sides of the left inequality givesdeg P k ≤ log q k + log q (deg P k −
1) + log q ( q −
1) + o (1) ≤ log q k + log q (log q k ) + log q ( q −
1) + o (1)where the upper bound for deg P k was substituted into the expression to obtain the second line.Likewise taking logs of the terms forming the right inequality of (5) givesdeg P k ≥ log q k + log q (deg P k ) + log q ( q − − o (1) ≥ log q k + log q (log q k ) + log q ( q − − o (1) . Again, the lower bound for deg P k was substituted into the expression to obtain the second line. (cid:3) Corollary 4.2. If { P k } is an arbitrary ordering of irreducible polynomials of increasing degree in M q , then deg P k = log q k + log q (log k ) + O (1) and k P k k ≍ q k log k. Remark . This result is essentially best possible, since we know that the degree of the k -thirreducible will jump by 1 every time the irreducibles of a given degree are exhausted (and thusthe norm will increase by a factor of q .) For comparison, over the integers it is known [7] thatlog p n = log n + log log n + log log n log n − n − (log log n ) − n + 52 log n + O (cid:18) log log n log n (cid:19) ! and that p n = n log n + log log n − n − n + O (cid:18) log log n log n (cid:19) !! where p n is the n -th prime number.5. Polynomials with k irreducible factors Having investigated the distribution of irreducibles in the function field, a natural extension is toconsider the distribution of monic polynomials with k irreducible factors. To this end, we introducethe counting function Π ′ q,k ( n ), which denotes the number of squarefree monic polynomials of degree n with k irreducible factors. Here, we investigate an asymptotic formula for Π ′ q,k ( n ) and determinea strict upper bound that gives us a quantitative analogue of the Hardy-Ramanujan theorem forfunction fields. .1. The Sathe-Selberg Formula.
Afshar and Porrit [1] recently showed that an analogue ofthe Sathe-Selberg theorem holds for function fields, which gives us an asymptotic expression forΠ ′ q,k ( n ). Theorem 5.1.
Let C ∈ [1 , . For n ≥ and ≤ k ≤ C log n + 1 , Π ′ q,k ( n ) = q n n · log k − n ( k − (cid:18) G (cid:18) k − n (cid:19) + O C (cid:18) k log n (cid:19)(cid:19) , where G ( z ) = 1Γ( z + 1) Y p ∈I q (cid:18) z k p k (cid:19) (cid:18) − k p k (cid:19) z . We can obtain a simplified version of this asymptotic by showing that G ( k − n ) is bounded awayfrom zero and infinity in this range. Lemma 5.2.
In the range ≤ z ≤ we have e ≤ G ( z ) ≤ . Proof.
Note that ddz (cid:18) z k p k (cid:19) (cid:18) − k p k (cid:19) z = (cid:18) − k p k (cid:19) z (cid:18)(cid:18) z k p k (cid:19) log (cid:18) − k p k (cid:19) + 1 k p k (cid:19) < z >
0. Since z +1) is also decreasing on this range, we can bound G (2) ≤ G ( z ) ≤ G (0) forall z in this ranhaveupper bound is then obtained by noting that G (0) = 1. We obtain the lowerbound by estimating G (2). G (2) = 1Γ(3) Y p ∈I q (cid:18) k p k (cid:19) (cid:18) − k p k (cid:19) = 12 ∞ Y n =1 (cid:18) q n (cid:19) (cid:18) − q n (cid:19) ! π ′ ( n ) ≥ ∞ Y n =1 (cid:18) − q n + 2 q n (cid:19) qnn . Taking logs, and using that log(1 − x ) ≥ − x − x for 0 < x ≤ / − log 2 + ∞ X n =1 q n n log (cid:18) − q n + 2 q n (cid:19) > − log 2 − ∞ X n =1 q n n (cid:18) q n − q n (cid:19) + (cid:18) q n − q n (cid:19) ! = − log 2 − ∞ X n =1 (cid:18) nq n − nq n + 9 nq n − nq n + 4 nq n (cid:19) = − log 2 + 3 log (cid:18) − q (cid:19) − (cid:18) − q (cid:19) + 9 log (cid:18) − q (cid:19) −
12 log (cid:18) − q (cid:19) + 4 log (cid:18) − q (cid:19) ≥ − log 2 + 3 log (cid:18) (cid:19) − (cid:18) (cid:19) + 9 log (cid:18) (cid:19) −
12 log (cid:18) (cid:19) + 4 log (cid:18) (cid:19) > − . G (2) > e − as claimed. (cid:3) Because G ( k − n ) is bounded away from zero, we can write our asymptotic for Π ′ q,k ( n ) in a moreconvenient form. If we define H k ( n ) = q n n · log k − n ( k − G (cid:18) k − n (cid:19) , hen we can write Theorem 5.1 in the formΠ ′ q,k ( n ) = H k ( n ) (cid:18) O C (cid:18) k log n (cid:19)(cid:19) , (6)which we will find useful for later sections.5.2. The Hardy-Ramanjuan Inequality.
The Sathe-Selberg formula implies thatΠ ′ q,k ( n ) = O q n log k − nn ( k − ! for all k ∈ [1 , C log n + 1]. In this section, we obtain a uniform upper bound of this form, validfor all k and n , and use it to obtain bounds for the number of monic polynomials with at most α log n or at least β log n prime factors. Our result can be interpreted as a quantitative version ofthe Hardy-Ramanujan theorem for function fields, that almost all degree n polynomials have aboutlog n distinct prime factors. Lemma 5.3. ( k − ′ q,k ( n ) ≤ X p ∈I q deg p ≤ n/ Π ′ q,k − ( n − deg p ) . Proof.
Each polynomial counted by Π ′ q,k ( n ) has the form p p . . . p k , where the p i are distinctirreducibles whose degrees add to n . At least k − p i have degree less than n/
2. If we fix anirreducible p with degree less than n/
2, we can multiply it with a squarefree polynomial of degree n − deg p with k − n polynomial with k irreducible factors.This polynomial may no longer be squarefree, but this is acceptable since we are only looking foran upper bound.For each choice of p , we obtain Π ′ q,k − ( n − deg p ) such polynomials with k prime factors. Summingover all irreducibles p of degree at most n/ X p ∈I q deg p ≤ n/ Π ′ q,k − ( n − deg p ) . This expression overcounts Π ′ q,k ( n ) by at least a factor of k −
1, since there are at least k − p was used to construct a given such polynomial with k prime factors. Hence,dividing it by k − ′ q,k ( n ). (cid:3) We now derive an explicit upper bound for the number of squarefree polynomials of degree n having exactly k factors. Similar, explicit bounds have been given over the integers, see for exampleTheorem 3.5 of [4]. Theorem 5.4. Π ′ q,k ( n ) ≤ q n n (log n + c ) k − ( k − for all k and n , where c = 2 − log 2 = 1 . . . . Proof.
We establish the claim by induction on k . When k = 1, Π ′ q,k ( n ) counts the number of monicirreducibles of degree n , so Π ′ ( n ) = π ′ q ( n ) ≤ q n n .Now assume the claim is true for k = j , so thatΠ ′ q,j ( n ) ≤ q n n (log n + c ) j − ( j − . hen by Lemma 5.3, Π ′ q,j +1 ( n ) ≤ j X p ∈I q deg p ≤ n/ q n − deg p n − deg p (log( n − deg p ) + c ) j − ( j − ≤ q n (log n + c ) j − j ! X p ∈I q deg p ≤ n/ n − deg p ) q deg p . (7)We now find an upper bound for this sum. Note that X p ∈I q deg p ≤ n/ n − deg p ) q deg p = X p ∈I q deg p ≤ n/ nq deg p · − deg pn = X p ∈I q deg p ≤ n/ nq deg p pn + (cid:18) deg pn (cid:19) + . . . ! = X p ∈I q deg p ≤ n/ (cid:18) nq deg p + deg pn q deg p (cid:18) pn + · · · (cid:19)(cid:19) = X p ∈I q deg p ≤ n/ nq deg p + X p ∈I q deg p ≤ n/ deg pn q deg p · − deg pn ! . We bound the first summation by noting that X p ∈I q deg p ≤ n/ nq deg p = 1 n ⌊ n/ ⌋ X k =1 π ′ q ( k ) q k ≤ n ⌊ n/ ⌋ X k =1 k ≤ log n − log 2 + 1 n . For the second summation, note that 1 / (1 − deg pn ) ≤ p ≤ n/
2, so X p ∈I q deg p ≤ n/ deg pn q deg p · − deg pn ! ≤ n ⌊ n/ ⌋ X k =1 kπ ′ q ( k ) q k ≤ n ⌊ n/ ⌋ X k =1 ≤ n · n n . Hence X p ∈I q deg p ≤ n/ n − deg p ) q deg p ≤ log n + 2 − log 2 n . Inserting this into (7) we can concludeΠ ′ q,j +1 ( n ) ≤ q n n (log n + c ) j j !as desired. (cid:3) Proposition 5.5.
Let n be an integer and let α and β be constants such that < α < < β . Thenthe number of polynomials of degree n having less than α log n or more than β log n prime factorssatisfies the following bounds. X k ≤ α log n Π ′ q,k ( n ) ≪ α q n n Q ( α ) √ log n and X k ≥ β log n Π ′ q,k ( n ) ≪ β q n n Q ( β ) √ log n , where Q ( y ) = y log y − y + 1 . roof. By Theorem 5.4, X k ≤ α log n Π ′ q,k ( n ) ≤ X k ≤⌊ α log n ⌋ q n (log n + c ) k − n ( k − X k ≥ β log n Π ′ q,k ( n ) ≤ X k ≥ β log n q n (log n + c ) k − n ( k − . When x > < α < < β , [13] gives us the bounds X k ≤ αx e − x x k k ! < e − Q ( α ) x (1 − α ) √ αx and X k ≥ βx e − x x k k ! < e − Q ( β ) x ( β − √ πβx , (8)where Q ( y ) = y log y − y + 1. Letting x = log n + c in the first expression gives us X k ≤ α log n Π ′ q,k ( n ) ≤ q n e c X k ≤ α (log n + c ) (log n + c ) k − e log n + c ( k − < q n e c − Q ( α )(log n + c ) (1 − α ) p α (log n + c ) ≪ α q n n Q ( α ) √ log n . For the second expression we have, using (8) and Stirling’s approximation X k ≥ β log n Π ′ q,k ( n ) = X β log n ≤ k<β (log n + c ) Π ′ q,k ( n ) + X k ≥ β (log n + c ) Π ′ q,k ( n ) < ( cβ + 1) q n (log n + c ) ⌊ β log n ⌋− n ( ⌊ β log n ⌋ − q n e c X k ≥ β (log n + c ) (log n + c ) k − e log n + c ( k − ≪ β q n (log n + c ) ⌊ β log n ⌋− n √ log n (( β log n − /e ) ⌊ β log n ⌋− + q n n Q ( β ) √ log n ≪ q n n √ log n (cid:18) eβ (cid:19) ⌊ β log n ⌋− + q n n Q ( β ) √ log n ≪ q n n Q ( β ) √ log n . (cid:3) Primitive Sets with Consistently Growing Counting Functions
Having considered the asymptotics of π ′ q ( n ) and Π ′ q,k ( n ), we are ready to construct primitive setswith consistently growing counting functions. We adapt a construction of Martin and Pomerancein the integers [12] to prove the following theorem. Theorem 6.1.
Suppose L ( x ) is positive and increasing, that L ( x ) ∼ L (2 x ) , and that Z ∞ dtt log t · L ( t ) < ∞ . Then there exists a primitive set S ⊂ M q such that S ′ ( n ) satisfies S ′ ( n ) ≍ q n log n · log log n · L (log n ) . Taking L ( x ) = log log x · log log log x · · · (cid:0) log j x (cid:1) ε in Theorem 6.1 for some j ≥ j ( x )denotes the j -fold iterated logarithm) we have the following corollary. Corollary 6.2.
For any ε > and j ≥ there exists a primitive set S ⊂ M q such that S ′ ( n ) satisfies S ′ ( n ) ≍ q n log n · log log n · · · (cid:0) log j n (cid:1) ε . In a sense, this is the slowest-growing function that L ( x ) could possibly be, since R ∞ dtt log t ··· (log j t ) ε grows without bound as ε tends to 0. Thus, our corollary gives the fastest-growing asymptoticcounting function achievable using Theorem 6.1. In particular, it is much larger than the countingfunction of the irreducible polynomials of degree n , which is asymptotic to q n n . .1. Constructing a sequence of irreducible polynomials.
A critical ingredient in the con-struction of our primitive set S will be a sequence of monic irreducible polynomials { t k } in M q .In order for S ′ ( n ) to have the desired asymptotics, we need to impose two conditions this sequence { t k } : P ∞ i =1 1 k t i k < and k t k k ≍ kL ( k ) log k . The following proposition guarantees the existence ofsuch a sequence. Proposition 6.3.
Let L ( x ) be positive and increasing, such that L ( x ) ∼ L (2 x ) and Z ∞ dtt log tL ( t ) < ∞ . Then there exists a sequence of irreducible polynomials { t i } of nondecreasing degree such that P ∞ i =1 1 k t i k < and k t k k ≍ q kL ( k ) log k .Proof. Because L ( x ) is increasing and R ∞ dttL ( t ) converges, there exists some integer y such that L ( y ) ≥ y ≥ y . Without loss of generality let y ≥
3. Fix an ordering of the irreduciblepolynomials that respects degree and consider the sequence of polynomials { r k } defined by r k = (cid:26) the k th irreducible polynomial if k < y the ⌊ kL ( k ) ⌋ th irreducible polynomial if k ≥ y . For k ≥ y we have that ⌊ ( k + 1) L ( k + 1) ⌋ ≥ ⌊ ( k + 1) L ( k ) ⌋ ≥ ⌊ kL ( k ) + 1 ⌋ = ⌊ kL ( k ) ⌋ + 1, so theindices of the polynomials in { r k } are strictly increasing for all k . By Corollary 4.2, the degree ofthe k th irreducible polynomial is log q ( k ) + log q (log k ) + O (1). Using this, we can find that when k ≥ y , k r k k = q log q ( kL ( k ))+log q (log( kL ( k )))+ O (1) = kL ( k ) log( kL ( k )) q O (1) ≍ q kL ( k ) log( k ) . Thus ∞ X k = y k r k k ≪ q ∞ X k = y kL ( k ) log k ≤ Z ∞ dtt log tL ( t ) . Since the last integral is bounded by assumption, the left hand sum must also be bounded. Thusthere exists some k ≥ y such that P ∞ k = k k r k k < .Then the sequence { t k } given by t k = r k + k has the property that P ∞ k =1 1 k t k k < . Furthermore,by Corollary 4.2, k t k k ≍ ⌊ ( k − k ) L ( k − k ) ⌋ log( ⌊ ( k − k ) L ( k − k ) ⌋ ) ∼ kL ( k ) log k. (cid:3) We now use the sequence { t k } from Proposition 6.3 to construct a primitive set by defining S k = { f ∈ M q : f squarefree, ω ( f ) = k, t k | f, and t j f for all j < k } , where ω ( f ) denotes the number of irreducible factors of f . Then, we let S = S ∞ k =1 S k . Proposition 6.4. S is a primitive set.Proof. Each S k is primitive, since any multiple of a polynomial with k irreducible factors must havemore than k irreducible factors. Hence if we assume for the sake of contradiction that S is notprimitive, then there exist f, g ∈ S with f | g such that f ∈ S m and g ∈ S n where m < n . Howeverthis is impossible since t n | f but t n g . (cid:3) Asymptotics of S ′ k ( n ) .Lemma 6.5. Suppose k ≪ log n and the sequence of polynomials { t k } is as defined above. Then k − ( n − deg t k ) = O (cid:18) n (cid:19) . Proof.
Since deg t k ≪ log k ≪ log log n , k − ( n − deg t k ) ≪ log log n log n ≪ n . (cid:3) ote that this result also holds if deg t k is replaced with deg t k + deg t j for some j < k becausedeg t k + deg t j = O (deg t k ). Theorem 6.6.
For sufficiently large n and k ∈ [2 , log n ] , S ′ k ( n ) ≍ q n − deg t k n − deg t k · log( n − deg t k ) k − ( k − . Proof.
We can bound S ′ k ( n ) by Π ′ q,k ( n ) asΠ ′ q,k − ( n − deg t k ) ≥ S ′ k ( n ) ≥ Π ′ q,k − ( n − deg t k ) − k − X j =1 Π ′ q,k − ( n − deg t j − deg t k ) . Here, the upper bound comes from the observation that S ′ k ( n ) counts a subset of the squarefreepolynomials f such that ω ( f ) = k and t k | f , while the lower bound is obtained by removing fromthis set the multiples of t j for all j < k . For sufficiently large n , we can apply Sathe-Selberg forfunction fields in the form of (6) to get S ′ k ( n ) ≤ H k − ( n − deg t k ) (cid:18) O (cid:18) k − ( n − deg t k ) (cid:19)(cid:19) = H k − ( n − deg t k ) O n !! , and S ′ k ( n ) ≥ H k − ( n − deg t k ) (cid:18) O (cid:18) k − ( n − deg t k ) (cid:19)(cid:19) − k − X j =1 H k − ( n − deg t k − deg t j ) O (cid:18) k − ( n − deg t k − deg t j ) (cid:19)! = H k − ( n − deg t k ) − k − X j =1 H k − ( n − deg t k − deg t j ) ! O n !! . (9)Here H k ( n ) is the function defined in Section 5.1, and Lemma 6.5 has been used to simplify theerror terms. Note that the upper bound already gives S ′ k ( n ) ≪ q n − deg t k n − deg t k · log( n − deg t k ) k − ( k − G ( z ) = O (1)) so we need only concern ourselves with the lower bound.We now factor out H k − ( n − deg t k ) from each term of (9) and consider the ratio P k − j =1 H k − ( n − deg t k − deg t j ) H k − ( n − deg t k )= ( k − k − X j =1 G (cid:16) k − n − deg t k − deg t j ) (cid:17) k t j k G (cid:16) k − n − deg t k ) (cid:17) n − deg t k n − deg t k − deg t j log k − ( n − deg t k − deg t j )log k − ( n − deg t k ) < k log( n − deg t k ) k − X j =1 G (cid:16) k − n − deg t k − deg t j ) (cid:17) k t j k G (cid:16) k − n − deg t k ) (cid:17) n − deg t k n − deg t k − deg t j = k log n k − X j =1 G (cid:16) k − n − deg t k − deg t j ) (cid:17) k t j k G (cid:16) k − n − deg t k ) (cid:17) (cid:18) O (cid:18) log nn (cid:19)(cid:19) . (10)Here we have used that n − deg t k = n + O (log k ) = n + O (log log n ) . ecause the function G ( z ), defined in Theorem 5.1, is analytic, and its derivative is bounded in theinterval [0 , G (cid:18) k − n − deg t k − deg t j ) (cid:19) = G (cid:18) k − n − deg t k ) + O (cid:18) n (cid:19)(cid:19) = G (cid:18) k − n − deg t k ) (cid:19) (cid:18) O (cid:18) n (cid:19)(cid:19) . Inserting this into (10) we find that P k − j =1 H k − ( n − deg t k − deg t j ) H k − ( n − deg t k ) ≤ k log n k − X j =1 k t j k (cid:18) O (cid:18) n (cid:19)(cid:19) . Using the expression above in (9), we see that S ′ k ( n ) ≥ H k − ( n − deg t k ) − k log n k − X j =1 k t j k ! O (cid:16) n (cid:17)! . Because we have chosen the t i with P k − j =1 1 k t j k < , and k ≤ log n we have 1 − k log n P k − j =1 1 k deg t j k > .This allows us to conclude that S ′ k ( n ) ≍ H k − ( n − deg t k ) ≍ q n − deg t k n − deg t k · log k − ( n − deg t k )( k − . (cid:3) The size of S ′ ( n ) . Proof of Theorem 6.1.
Let B = B ( n ) = ⌊ log n ⌋ and B ′ = B ′ ( n ) = ⌊ log n ⌋ . When n ≥ deg t ,we show that q n − deg t B ≫ S ′ ( n ) ≫ q n − deg t B ′ . (11)Because S is a disjoint union of the sets S k , we can bound S ′ ( n ) using our bounds for S ′ k ( n ) fromTheorem 6.6. As a lower bound, we have S ′ ( n ) ≥ B ′ X k =1 S ′ k ( n ) ≫ B ′ X k =1 q n − deg t k n − deg t k · log k − ( n − deg t k )( k − ≥ q n − deg t B ′ n B ′ X k =1 log k − ( n − deg t B ′ )( k − . Since P ⌊ y ⌋ j =0 y j j ! ≫ e y , we get B ′ X k =2 log k − ( n − deg t B ′ )( k − ≫ n − deg t B ′ , which implies S ′ ( n ) ≫ q n − deg t B ′ . Similarly, we can bound S ′ ( n ) from above: S ′ ( n ) ≤ ∞ X k =1 S ′ k ( n ) ≤ B ′ X k = B +1 S ′ k ( n ) + X k ≤⌊ / n ⌋ Π ′ q,k ( n ) + X k ≥⌊ / n ⌋ Π ′ q,k ( n ) . From Proposition 5.5, we know that the latter two sums are bounded by O (cid:16) q n n Q (1 / (cid:17) and O (cid:16) q n n Q (3 / (cid:17) respectively, where Q ( ) , Q ( ) >
0. We can bound the first sum by Theorem 6.6: B ′ X k = B +1 S ′ k ( n ) ≪ B ′ X k = B +1 q n − deg t k n − deg t k log k − n ( k − ≤ q n − deg t B n − deg t B ∞ X j =0 log j nj ! log n = q m − deg t B n − deg t B ≤ q n − deg t B . Since this term grows faster than O ( q n n c ) for any positive constant c , we can disregard the other twosums and conclude that S ′ ( n ) ≪ q n − deg t B .Now, recall that our sequence of polynomials { t k } satisfies k t k k ≍ kL ( k ) log k ; etting k = c log n , we have k t k k ≍ c log n · L ( c log n ) · log( c log n ) ∼ c log n · log log n · L (log n ) . From (11) we have S ′ ( n ) ≫ q n − deg t B = q n k t ⌊ log n ⌋ k ≫ q n log n · log log n · L (log n ) , and similarly S ′ ( n ) ≪ q n − deg t B ′ = q n k t ⌊ log n ⌋ k ≪ q n log n · log log n · L (log n ) , proving the theorem. (cid:3) Future Research
Over the integers the Erd˝os sum (1) has been used as a metric to compare the relative size ofprimitive sets. In 1988, Erd˝os proposed that the primes are, in a sense, the “largest” primitive setunder this metric. In particular, he made the following conjecture, which has attracted significantrecent interest [3, 9, 11, 15] but remains open:
Conjecture 7.1. (Erd˝os) Let A ⊂ N be a primitive set, A = { } and P be the primes. Then, X a ∈ A a log a ≤ X p ∈P p log p We believe that the analogous conjecture holds in the function field case.
Conjecture 7.2.
Let A ⊂ M q be a primitive set, A = { } and I q be the irreducible polynomials.Then, X a ∈ A k a k deg a ≤ X f ∈I q k f k deg f . We will further investigate the size of this sum for primitive subsets of F q [ x ] in a future paper. Acknowledgements
This research was conducted as part of the SUMRY (Summer Undergraduate Mathematics Re-search at Yale) program in Summer 2019. We would also like to thank the anonymous referee forhelpful suggestions.
References [1] A. Afshar and S. Porritt,
The function field Sathe-Selberg formula in arithmetic progressionsand ‘short intervals’ , Acta Arith. (2019), no. 2, 101–124. MR3897499[2] R. Ahlswede, L. H. Khachatrian, and A. S´ark¨ozy,
On the counting function of primitive setsof integers , J. Number Theory (1999), no. 2, 330–344. MR1729755[3] W. D. Banks and G. Martin, Optimal primitive sets with restricted primes , Integers (2013),Paper No. A69, 10. MR3118387[4] J. Bayless, P. Kinlaw, and D. Klyve, Sums over primitive sets with a fixed number of primefactors , Math. Comp. (2019), no. 320, 3063–3077. MR3985487[5] A. S. Besicovitch, On the density of certain sequences of integers , Math. Ann. (1935),no. 1, 336–341. MR1512943[6] M. Car,
Polynˆomes de F q [ X ] ayant un diviseur de degr´e donn´e , Acta Arith. (1984), no. 2,131–154. MR736727[7] M. Cipolla, La determinazione assintotica dell’ n imo numero primo , Rendiconto dell’Accademiadelle Scienze Fisiche e Matematiche, Napoli (1902), 132–166.
8] P. Erd˝os,
Note on sequences of integers no one of which is divisible by any other , J. LondonMath. Soc. (1935), no. 2, 126–128. MR1574239[9] P. Erd˝os and Z. X. Zhang, Upper bound of P / ( a i log a i ) for primitive sequences , Proc. Amer.Math. Soc. (1993), no. 4, 891–895. MR1116257[10] M. Kruse and H. Stichtenoth, Ein Analogon zum Primzahlsatz f¨ur algebraische Funktio-nenk¨orper , Manuscripta Math. (1990), no. 3, 219–221. MR1078353[11] J. D. Lichtman and C. Pomerance, The Erd˝os conjecture for primitive sets , Proc. Amer. Math.Soc. Ser. B (2019), 1–14. MR3937344[12] G. Martin and C. Pomerance, Primitive sets with large counting functions , Publ. Math. De-brecen (2011), no. 3-4, 521–530. MR2907985[13] K. K. Norton, On the number of restricted prime factors of an integer. I , Illinois J. Math. (1976), no. 4, 681–705. MR0419382[14] M. Rosen, A generalization of Mertens’ theorem , J. Ramanujan Math. Soc. (1999), no. 1,1–19. MR1700882[15] Z. X. Zhang, On a conjecture of Erd˝os on the sum P p ≤ n / ( p log p ), J. Number Theory (1991), no. 1, 14–17. MR1123165 Department of Mathematics, Yale University, New Haven, CT 06520
E-mail address : [email protected] Department of Mathematics, Yale University, New Haven, CT 06520
E-mail address : [email protected] Department of Mathematics, Towson University, Towson, MD 21252
E-mail address : [email protected] Department of Mathematics, Yale University, New Haven, CT 06520
E-mail address : [email protected]@yale.edu