Erdős Primitive Set Conjecture Resolved

• The Erdős Primitive Set Conjecture defines primitive sets as those where no element divides another, establishing primes as yielding the maximal Erdős sum.
• Advanced techniques such as density-of-multiples sieves and von Mangoldt Markov chains were key in proving that primes optimize the sum.
• Its resolution has spurred further research in extremal number theory, influencing studies on prime divisor bounds, weighted sums, and k-primitive generalizations.

A primitive set in the positive integers is one in which no element divides another; the concept is central in understanding extremal questions concerning multiplicative structure within subsets of ℕ. The Erdős Primitive Set Conjecture, formulated by Paul Erdős in 1986, asserts that among all such primitive sets, the set of prime numbers maximizes the sum

$S(A) = \sum_{a \in A} \frac{1}{a \log a}\,.$

This sum is known as the "Erdős sum" (Editor's term), and the conjecture further asserts its maximal value is

$$S(\mathbb{P}) = \sum_{p\text{ prime}} \frac{1}{p \log p} \approx 1.6366\,.$$

The conjecture was settled affirmatively in 2022 by Lichtman and has since motivated a broad development of structural, analytic, and combinatorial methods in number theory.

1. Historical Development and Context

Primitive sets were introduced in 1935 by Erdős, who first proved that the sum $S(A)$ converges and is bounded by an absolute constant for every primitive set $A$ (Lichtman et al., 2018). Early bounds on $\sup S(A)$ were improved incrementally, with Erdős–Zhang, Robin, and Clark reducing the upper bound to $1.84$, $2.77$, and further, but the extremal structure remained elusive. In 1986, motivated by heuristic and numerical evidence, Erdős conjectured that the maximum is achieved precisely by the set of the primes.

Significant partial progress included the proof by Banks and Martin that for restricted primitive sets—those composed only of integers with all prime factors in a fixed subset of the primes—certain optimality properties hold, provided the base set of primes is sufficiently sparse (Banks et al., 2013). On the other hand, generalizations to $k$-primitive sets (no term divides the product of $k$ others) culminated in a definitive result by Chan, Lichtman, and Pomerance for $k = 2$, but fell short of the original ($$S(\mathbb{P}) = \sum_{p\text{ prime}} \frac{1}{p \log p} \approx 1.6366\,.$$0) conjecture (Chan et al., 2020).

The critical threshold and subtleties were further illuminated by connections to prime number races and sieve-theoretic methods, notably involving density-based and Mertens-product arguments (Lichtman et al., 2018).

2. Statement and Reformulations of the Conjecture

Let $$S(\mathbb{P}) = \sum_{p\text{ prime}} \frac{1}{p \log p} \approx 1.6366\,.$$1 be primitive: $$S(\mathbb{P}) = \sum_{p\text{ prime}} \frac{1}{p \log p} \approx 1.6366\,.$$2, $$S(\mathbb{P}) = \sum_{p\text{ prime}} \frac{1}{p \log p} \approx 1.6366\,.$$3 implies $$S(\mathbb{P}) = \sum_{p\text{ prime}} \frac{1}{p \log p} \approx 1.6366\,.$$4. The Erdős Primitive Set Conjecture states: $$S(\mathbb{P}) = \sum_{p\text{ prime}} \frac{1}{p \log p} \approx 1.6366\,.$$5 where $$S(\mathbb{P}) = \sum_{p\text{ prime}} \frac{1}{p \log p} \approx 1.6366\,.$$6 is the set of all prime numbers (Farhi, 2017, Lichtman, 2022).

Two reformulations are well-established and equivalent:

• Prime divisor bound: $$S(\mathbb{P}) = \sum_{p\text{ prime}} \frac{1}{p \log p} \approx 1.6366\,.$$7, where $$S(\mathbb{P}) = \sum_{p\text{ prime}} \frac{1}{p \log p} \approx 1.6366\,.$$8 is the set of prime divisors of elements of $$S(\mathbb{P}) = \sum_{p\text{ prime}} \frac{1}{p \log p} \approx 1.6366\,.$$9.
• Finite/rank-by-rank bound: For any finite primitive sequence $S(A)$0,

$S(A)$1

with $S(A)$2 the $S(A)$3th prime (Farhi, 2017).

The conjecture fails for certain weighted analogues; for the sum $S(A)$4, the "primes-are-best" property fails whenever $S(A)$5 (Farhi, 2017).

3. Proofs, Key Techniques, and Analytic Tools

The proof by Lichtman refines the classical "density-of-multiples" sieve technique due to Erdős (Lichtman, 2022). The main innovation is extracting a uniform "density saving" from composite elements, employing:

• The partitioning of $S(A)$6 by smallest prime factor $S(A)$7;
• Constructing for each $S(A)$8 the set $S(A)$9 and showing these are disjoint for primitive $A$0;
• Quantifying the saving via the ratio $A$1 for composites, together with disjointness among dilated $A$2 sets;
• Integrating these savings over suitable parameterizations yields an upper bound matching $A$3.

Key results rest on:

• Analytic properties of Mertens' product $A$4;
• Explicit Abel summation and density integration;
• Numerical bounds for small primes, ensuring the argument holds globally.

A parallel probabilistic/combinatorial approach—now yielding also remarkably short proofs—uses von Mangoldt Markov chains to derive hitting-mass inequalities for primitive sets, identifying the relevant weights as sub-invariant and establishing sharp upper bounds (Alexeev et al., 1 May 2026).

4. Extensions, Generalizations, and Related Problems

4.1 Restricted and Weighted Sums

Banks and Martin introduced the concept of a set of primes $A$5 being "Erdős-best" among primitive subsets of $A$6—the semigroup generated by $A$7—if

$A$8

for every primitive $A$9 (Banks et al., 2013).

They demonstrated that "Erdős-best" holds whenever $\sup S(A)$0 is sufficiently small, and gave a criterion for weighted sums $\sup S(A)$1 for $\sup S(A)$2, showing a phase transition at $\sup S(A)$3—a key boundary for the effectiveness of analytic techniques.

4.2 Higher $\sup S(A)$4-Primitive Sets

For $\sup S(A)$5-primitive sets (no element divides the product of $\sup S(A)$6 others), the maximal sum is always attained by the primes when $\sup S(A)$7, as proved by Chan, Lichtman, and Pomerance (Chan et al., 2020). This result crucially relies on combinatorial decompositions and matching lemmas, particularly for $\sup S(A)$8; however, techniques do not extend to $\sup S(A)$9 due to a critical-exponent obstruction.

4.3 Function Field Analogues

In the polynomial ring $1.84$0, the direct analogue considers primitive sets of monic polynomials (no element divides another). Analogous bounds have been established, with the conjecture that the maximal sum is attained by the set of monic irreducible polynomials. For $1.84$1, the exact Banks–Martin monotonicity analogue fails, but computational evidence indicates monotonicity for $1.84$2 (Gómez-Colunga et al., 2020).

5. Implications, Open Questions, and Future Directions

The definitive solution of the Erdős conjecture influences several directions:

• Ramsey-theoretic divisibility chains: The structural findings on L-multiples and chain decompositions suggest new problems in multiplicative posets (Lichtman, 2022).
• Quantitative improvements: The precise rate at which the supremum for primitive sets in intervals $1.84$3 converges to $1.84$4 as $1.84$5 remains of interest.
• Erdős-strong and Mertens/Zhang primes: The local version—whether every prime is "Erdős-strong" (optimal for smallest-prime-factor partitioned sets)—remains open in full generality, though all odd primes up to $1.84$6 have been proved so unconditionally, and under RH+LI almost all primes are of this kind (Lichtman et al., 2018).
• Weighted-sum critical exponents: Whether the phase transition $1.84$7 for sums $1.84$8 can be brought down to $1.84$9 in the general setting is open (Banks et al., 2013).
• Generalized weightings: The failure of the naive "primes are best" principle for translates $2.77$0 with $2.77$1 (Farhi, 2017) highlights the sharpness of the classical conjecture and motivates further study of extremal structures for other analytic functionals.

6. Summary Table of Landmark Results

Problem/Setting	Key Result	Reference
Original Erdős primitive set conjecture	Resolved affirmatively	(Lichtman, 2022)
k-primitive sets, $2.77$2	Primes maximize sum	(Chan et al., 2020)
Restricted primes ($2.77$3 Erdős-best)	Sufficient sparsity ⇒ optimal	(Banks et al., 2013)
Function field ($2.77$4)	Uniform upper bound; max by irreducibles (conj.)	(Gómez-Colunga et al., 2020)
Translated sums $2.77$5	Primes not always optimal	(Farhi, 2017)

The resolution of the conjecture has introduced new combinatorial and analytic methodologies, leading to advances in extremal number theory and stimulating a series of open problems at the interface of combinatorics, analytic number theory, and probability.