Do all power-moments of gaps between consecutive squarefree numbers converge? (Erdős #145)
Statement
Let $s_1<s_2<\cdots$ be the increasing sequence of squarefree numbers ($1,2,3,5,6,7,10,\ldots$). Is it true that for every real $\alpha\ge 0$ the limit $$\lim_{x\to\infty}\frac{1}{x}\sum_{s_n\le x}(s_{n+1}-s_n)^{\alpha}$$ exists?
Acceptance. FULLY RESOLVES: prove that the stated limit exists for every $\alpha\ge 0$, with a complete proof — a Lean/Coq formalisation is preferred, else a full written proof. An unconditional proof is the target; because the full statement is already known under abc [Gr98], a merely abc-conditional argument counts as fully resolving only if it adds genuinely new content. ADVANCES: prove existence of the limit for all $\alpha$ up to a threshold strictly larger than the best threshold stated in the background (currently $\alpha\le 3.75$), with proof; or establish a new unconditional upper bound on the extremal squarefree gaps that feeds this moment problem. Deliver the proof file specifying the exact range of $\alpha$ covered, or the improved gap bound with proof.
Background
Asked by Erdős [Er65b; Er79; Er81h, p.176]. Listed as open on erdosproblems.com/145 (fetched 2026-07-13, status 'open', tagged 'number theory'); a Lean 4 formalisation exists in google-deepmind/formal-conjectures. The squarefree numbers have density $1/\zeta(2)=6/\pi^2$, so $s_n/n\to\zeta(2)=\pi^2/6$ and the mean gap is $\pi^2/6$; thus the $\alpha=1$ moment is classical; the content is the existence of the limiting average of ALL higher powers of the gaps. Known unconditional cases, by increasing difficulty in $\alpha$: Erdős [Er51] settled $0\le\alpha\le 2$; Hooley [Ho73] reached $\alpha\le 3$; Greaves, Harman and Huxley (Chapter 11 of [GHH97]) reached $\alpha\le 11/3$; and Chan [Ch23c] currently holds the record $\alpha\le 3.75$. Granville [Gr98] proved the full statement (all $\alpha\ge 0$) as a consequence of the abc conjecture. The obstruction is control of large gaps between squarefree numbers — see the companion Erdős #208 (erdosproblems.com/208) on the maximal gap $s_{n+1}-s_n$. No Erdős prize is recorded. Attacker's tool: sieve and analytic number theory extending the Filaseta–Trifonov-type technology behind the current $\alpha\le 3.75$ threshold, supported by large-scale computation of the empirical moments (via a segmented squarefree sieve) to test convergence and calibrate the extremal-gap input.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Erdős Problem #145 (T. F. Bloom) | website |
| REF-02 | OEIS A005117 — the squarefree numbers | website |
| REF-03 | Lean 4 formalisation — google-deepmind/formal-conjectures #145 | website |
Investigations · 0
No published investigations yet. This problem is unclaimed territory.