The Erdős conjecture for primitive sets

By Jared Duker Lichtman and Carl Pomerance

Abstract

A subset of the integers larger than 1 is primitive if no member divides another. Erdős proved in 1935 that the sum of for running over a primitive set is universally bounded over all choices for . In 1988 he asked if this universal bound is attained for the set of prime numbers. In this paper we make some progress on several fronts and show a connection to certain prime number “races” such as the race between and .

1. Introduction

A set of positive integers is called primitive if no element divides any other (for convenience, we exclude the singleton set ). There are a number of interesting and sometimes unexpected theorems about primitive sets. After Besicovitch Reference 4, we know that the upper asymptotic density of a primitive set can be arbitrarily close to , whereas the lower asymptotic density is always . Using the fact that if a primitive set has a finite reciprocal sum, then the set of multiples of members of the set has an asymptotic density, Erdős gave an elementary proof that the set of nondeficient numbers (i.e., , where is the sum-of-divisors function) has an asymptotic density. Though the reciprocal sum of a primitive set can possibly diverge, Erdős Reference 9 showed that for a primitive set ,

In fact, the proof shows that these sums are uniformly bounded as varies over primitive sets.

Some years later in a 1988 seminar in Limoges, Erdős suggested that in fact we always have

where is the set of prime numbers. The assertion Equation 1.1 is now known as the Erdős conjecture for primitive sets.

In 1991, Zhang Reference 19 proved the Erdős conjecture for primitive sets with no member having more than 4 prime factors (counted with multiplicity).

After Cohen Reference 6, we have

the sum over primes in Equation 1.1. Using the original Erdős argument in Reference 9, Erdős and Zhang showed that for a primitive set , which was later improved by Robin to . These unpublished estimates are reported in Reference 11 by Erdős–Zhang, who used another method to show that . Shortly after, Clark Reference 5 claimed that  . However, his brief argument appears to be incomplete.

Our principal results are the following.

Theorem 1.1.

For any primitive set we have .

Theorem 1.2.

For any primitive set with no element divisible by , we have .

Say a prime is Erdős strong if for any primitive set with the property that each element of has the same least prime factor , we have . We conjecture that every prime is Erdős strong. Note that the Erdős conjecture Equation 1.1 would immediately follow, though it is not clear that the Erdős conjecture implies our conjecture. Just proving our conjecture for the case of would give the inequality in Theorem 1.2 for all primitive sets . Currently the best we can do for a primitive set of even numbers is that ; see Proposition 2.1 below.

For part of the next result, we assume the Riemann Hypothesis (RH) and the Linear Independence Hypothesis (LI), which asserts that the sequence of numbers such that is linearly independent over .

Theorem 1.3.

Unconditionally, all of the odd primes among the first primes are Erdős strong. Assuming RH and LI, the Erdős strong primes have relative lower logarithmic density .

The proof depends strongly on a recent result of Lamzouri Reference 13, who was interested in the “Mertens race” between and .

For a primitive set , let denote the support of , i.e., the set of prime numbers that divide some member of . It is clear that the Erdős conjecture Equation 1.1 is equivalent to the same assertion where the prime sum is over .

Theorem 1.4.

If is a primitive set with , then

If some primitive set of odd numbers exists with , Theorem 1.4 suggests that it will be very difficult indeed to give a concrete example!

For a positive integer , let denote the number of prime factors of counted with multiplicity. Let denote the set of integers with . Zhang Reference 20 proved a result that implies for each , so that the Erdős conjecture holds for the primitive sets . More recently, Banks and Martin Reference 2 conjectured that  . The inequality was just established by Bayless, Kinlaw, and Klyve Reference 3. We prove the following result.

Theorem 1.5.

There is a positive constant such that for all .

We let the letters represent primes. In addition, we let represent the th prime. For an integer , we let and denote the largest and smallest prime factors of . Modifying the notation introduced in Reference 11, for a primitive set let

We let , and so . In this language, Zhang’s full result Reference 20 states that for all primes , . We also let

with and .

2. The Erdős approach

In this section we will prove Theorem 1.1. We begin with an argument inspired by the original 1935 paper of Erdős Reference 9.

Proposition 2.1.

For any primitive set , if , then

Proof.

For each , let . Note that has asymptotic density . Since is primitive, we see that the sets are pairwise disjoint. Further, the union of the sets is contained in the set of all natural numbers with , which has asymptotic density . Thus, the sum of densities for each is dominated by , that is,

By Theorem 7 in Reference 17, we have for ,

which may be extended to all by a calculation. Thus, since each is composite,

Hence by Equation 2.1,

Remark 2.2.

Let denote the sum-of-divisors function, and let be the set of with and for all proper divisors of , the set of primitive nondeficient numbers. Then an appropriate analog of gives the density of nondeficient numbers recently shown in Reference 12 to lie in the tight interval . In Reference 14, an analog of Proposition 2.1 is a key ingredient for sharp bounds on the reciprocal sum of the primitive nondeficient numbers.

Remark 2.3.

We have . Indeed, it is easy to see by induction over primes that

Letting we get that . There is also a holistic way of seeing this. Since is the density of the set of integers with least prime factor , it would make sense that is the density of the set of integers which have a least prime factor, and this density is 1. To make this rigorous, one notes that the density of the set of integers whose least prime factor is tends to 0 as . As a consequence of , we have

an identity we will find to be useful.

For a primitive set , let

The next result will help us prove Theorem 1.1.

Lemma 2.4.

For a primitive set , let be such that . Then we have

Proof.

The hypothesis implies that , so that is a primitive set. If for a prime , then is a primitive set of odd composite numbers, so by Proposition 2.1, .

Now if for some odd prime , then , and note that by primitivity. We have since

which follows from Equation 2.2. Since implies , we have

With Lemma 2.4 in hand, we prove .

Proof of Theorem 1.1.

From Erdős–Zhang Reference 11, we have that . If , then , so that . Hence we may assume that . If contains every odd prime, then consists of at most one power of 2, and the calculation just concluded shows we may assume this is not the case. Hence there is at least one odd prime . By Proposition 2.1, we have

First suppose contains no powers of . Then by Lemma 2.4,

Substituting into Equation 2.4, we conclude, using Equation 2.3, that

For the last inequality in Equation 2.5 we used that for every prime ,

which follows after a short calculation using Reference 17, Theorem 7.

Now if for some positive integer , then is unique and . Also and for all , so again by Lemma 2.4,

Substituting into Equation 2.4 gives

using identity Equation 2.3, inequality Equation 2.6, and . This completes the proof.

3. Mertens primes

In this section we will prove Theorems 1.3 and 1.4. Note that by Mertens’ theorem,

where is Euler’s constant. We say a prime is Mertens if

and let denote the set of Mertens primes. We are interested in Mertens primes because of the following consequence of Proposition 2.1, which shows that every Mertens prime is Erdős strong.

Corollary 3.1.

Let be a primitive set. If , then . Hence if for all , then satisfies the Erdős conjecture.

Proof.

By Proposition 2.1 we have . If , then

so .

Now, one would hope that the Mertens inequality Equation 3.1 holds for all primes . However, Equation 3.1 fails for since . Nevertheless, we have computed that is indeed a Mertens prime for all , thus proving the unconditional part of Theorem 1.3.

3.1. Proof of Theorem 1.3

To complete the proof, we use a result of Lamzouri Reference 13 relating the Mertens inequality to the race between and , studied by Rubinstein and Sarnak Reference 18. Under the assumption of RH and LI, he proved that the set of real numbers satisfying

has logarithmic density equal to the logarithmic density of numbers with , and in particular

We note that if a prime , then for