An Average-Order Theorem for a Shifted Pairwise-Coprime Extremal Problem

Abstract: For $n\ge 2$, let $\mathcal{M}(n)$ be the supremum of $\sum_{a\in A}1/(n-a)$ over pairwise coprime sets $A\subset [1,n)$. Erdős asked whether $\mathcal{M}(n)\le \sum_{p<n}1/p+O(1)$ uniformly in $n$. We prove the quantitative average-order formula $$\sum_{n\le N}\mathcal{M}(n) = e^{-γ}N\log\log N+O(N).$$ The lower bound comes from the self-rough construction $\{n-d:P^{-}(n-d)>d}$, while the upper bound uses bounded-cost dual certificates and Buchstab--de Bruijn estimates for rough numbers. We also prove that $\mathcal{M}(n)=(e^{{-γ}+o(1))\log\log} n$ for almost all $n$, with a quantitative exceptional-set bound, and hence Erdős's inequality holds for almost all $n$. The almost-all proof uses a long-interval two-dimensional beta-sieve estimate for two moving forbidden residue classes, together with an exact finite singular-series cancellation. Finally, we prove the pointwise bound $$\mathcal{M}(n)\le (2+\varepsilon)\log\log n+O_{\varepsilon}(1)$$, explain the linear-sieve barrier behind the constant $2$, and record structural certificates, conditional window-packing reductions, numerical examples, and CRT sharpness constructions.

• The paper establishes the average-order formula Σₙ₍ₙ₎ = e^(–γ) N log log N + O(N), advancing Erdős’s extremal problem.
• It applies a dual certificate and refined sieve techniques, including dyadic decompositions and beta-sieve methods, to obtain sharp bounds.
• The study quantifies almost-everywhere behavior and structural limitations, setting a foundation for further work in multiplicative combinatorics.

Average-Order Analysis of a Shifted Pairwise-Coprime Extremal Problem

Problem Statement and Historical Background

The paper "An Average-Order Theorem for a Shifted Pairwise-Coprime Extremal Problem" (2606.17955) addresses a longstanding question posed by Erdős regarding extremal properties of sets of pairwise coprime integers. Specifically, for $n \geq 2$, the author studies the maximal sum

$$(n) = \sup_{\substack{A \subset [1,n) \ (a,b) = 1,\, a \neq b}} \sum_{a \in A} \frac{1}{n - a},$$

where $A$ ranges over all pairwise coprime subsets of $\{1,2,\ldots,n-1\}$. Erdős conjectured that

$(n) \leq \sum_{p < n} \frac{1}{p} + O(1)$

with a uniform constant, where the sum is over primes $p < n$.

This extremal problem integrates several themes in analytic number theory: distribution of pairwise coprime sets, local properties amplified by short intervals (e.g., shifted primes), and techniques from sieve theory. The complexity is further accentuated by the unavoidable presence of shifted-prime reciprocals, reflecting the sharpness of the extremal construction.

Main Results

Average-Order Theorem

The core achievement is a precise asymptotic formula for the average value of $(n)$:

$$\sum_{n \leq N} (n) = e^{-\gamma} N \log\log N + O(N)$$

where $e^{-\gamma}$ is the Buchstab constant, with $\gamma$ the Euler-Mascheroni constant. This average order is strictly smaller, by a positive proportion, than the trivial sum over shifted prime reciprocals. The result answers, in the mean, Erdős's question, proving that the average maximal sum is parametrically less than the corresponding prime harmonic sum.

The method is robust: the lower bound is attained via a "self-rough" construction taking $$(n) = \sup_{\substack{A \subset [1,n) \ (a,b) = 1,\, a \neq b}} \sum_{a \in A} \frac{1}{n - a},$$0, which exploits the abundance of $$(n) = \sup_{\substack{A \subset [1,n) \ (a,b) = 1,\, a \neq b}} \sum_{a \in A} \frac{1}{n - a},$$1-rough numbers, and the upper bound employs a dual certificate and precise estimates for the count of rough numbers, leveraging sieve techniques and the Buchstab--de Bruijn theorem.

Almost-Everywhere Behavior

The paper further establishes that for almost all $$(n) = \sup_{\substack{A \subset [1,n) \ (a,b) = 1,\, a \neq b}} \sum_{a \in A} \frac{1}{n - a},$$2,

$$(n) = \sup_{\substack{A \subset [1,n) \ (a,b) = 1,\, a \neq b}} \sum_{a \in A} \frac{1}{n - a},$$3

i.e., the extremal sum is concentrated at the Buchstab value on the logarithmic scale. Quantitatively, for any $$(n) = \sup_{\substack{A \subset [1,n) \ (a,b) = 1,\, a \neq b}} \sum_{a \in A} \frac{1}{n - a},$$4,

$$(n) = \sup_{\substack{A \subset [1,n) \ (a,b) = 1,\, a \neq b}} \sum_{a \in A} \frac{1}{n - a},$$5

Notably, this almost-everywhere result implies Erdős’s original inequality holds for all sufficiently large $$(n) = \sup_{\substack{A \subset [1,n) \ (a,b) = 1,\, a \neq b}} \sum_{a \in A} \frac{1}{n - a},$$6 outside a zero-density exceptional set.

The method for almost-everywhere results hinges on a detailed analysis of the variance of $$(n) = \sup_{\substack{A \subset [1,n) \ (a,b) = 1,\, a \neq b}} \sum_{a \in A} \frac{1}{n - a},$$7 using a two-dimensional beta-sieve, singular-series cancellation, and a careful partition between diagonal and off-diagonal contributions. The analysis makes essential use of the explicitly quantified parity structure in the singular series, offsetting large positive and negative covariance terms in the fluctuation of the sum.

Technical Innovations

The upper bound is established by a novel "^^^^2^^^^" dual certificate, combining:

• Dyadic decompositions and the linear Rosser–Iwaniec sieve for intervals.
• Crude upper-bound sieves for short intervals.
• Explicit singular-series computations and mean-value identities, notably an exact finite mean value for local correlation factors.
• A two-point (dimension-two) beta-sieve result over long intervals with varying residue classes, critical for the two-dimensional variance estimate.

For the variance estimate, the paper makes an explicit decomposition of correlations, separating even and odd gaps, and introduces a finite-product singular series to capture off-diagonal covariance. Parity is exploited exactly (not up to absolute value), which is essential for achieving the cancellation necessary for the correct order of the second moment.

Pointwise and Exceptional-Set Results

While the main theorems are average and "almost all" statements, the paper also provides pointwise upper bounds:

$$(n) = \sup_{\substack{A \subset [1,n) \ (a,b) = 1,\, a \neq b}} \sum_{a \in A} \frac{1}{n - a},$$8

for any $$(n) = \sup_{\substack{A \subset [1,n) \ (a,b) = 1,\, a \neq b}} \sum_{a \in A} \frac{1}{n - a},$$9. This bound is shown to be sharp under the linear sieve alone—the "2" arises from the sieve barrier at $A$0. Any improvement to constant $A$1 would require input comparable to a Hardy–Littlewood type short-interval prime counting result.

The structural limitations are discussed in the context of window-packing and exchange lemmas, and the precise local combinatorial certificate necessary for the conjectured bound is isolated as a hypothesis strictly stronger than existing large sieve or linear sieve approaches.

Implications for Extreme Constructions

The paper develops a range of structural and probabilistic constructions (e.g., using the Chinese Remainder Theorem) to probe the tightness of available upper bounds, showing in particular that simple strategies (like replacing every composite by a nearby prime divisor in the sum) cannot attain the conjectural constant.

Moreover, the extremal sets built are shown, via local packing certificates, to be highly dependent on the fine distribution of prime numbers in short intervals, thus connecting the extremal problem directly to unresolved conjectures in prime number theory.

Future Directions

• Variance and Limiting Laws: The current work achieves full average and normal order results, but $A$2 variance for the untruncated sum remains open. Achieving such variance would likely necessitate establishing finite-$A$3 two-dimensional Buchstab estimates for larger sieve levels and handling shifted-prime contributions in the tail.
• Pointwise Bounds: Sharpening the constant in the pointwise bound and identifying explicit constructions (or obstructions) for the limsup of $A$4 remain open, with connections to short-interval prime gaps and Hardy–Littlewood majorant properties.
• Erdős--Kac Type Law: It is natural to inquire about the limiting distribution (e.g., Gaussianity) of centered/normalized versions of $A$5 or its truncations, which the paper frames as open questions on higher moments and cumulants.
• Conditional Results: The paper precisely identifies window-packing hypotheses (akin to strengthened prime-interval theorems) that would imply the conjectural constant holds pointwise and notes their relation to classical and conditional results in sieve theory and the Hardy–Littlewood framework.

Conclusion

This work rigorously resolves, on average and for almost all $A$6, the extremal shifted pairwise-coprime reciprocal sum problem proposed by Erdős, achieving an exact leading constant. The juxtaposition of direct sieve-theoretic constructions with a detailed analysis of the variance structure and local combinatorics situates the paper at the interface of additive combinatorics, classical analytic number theory, and extremal set theory. The approach sharply distinguishes mean and pointwise phenomena, quantifies obstructions, and lays clear foundations for further study of extremal, probabilistic, and distributional properties in multiplicative combinatorics.