Prove $\sum_{n\le N}(p_{n+1}-p_n)^2\ll N(\log N)^2$ for prime gaps (Erdős #233)
Statement
Let $p_n$ denote the $n$th prime and let $d_n=p_{n+1}-p_n$ be the $n$th prime gap. Prove that $$\sum_{1\le n\le N}d_n^2\ll N(\log N)^2,$$ i.e. that there is an absolute constant $C$ with $\sum_{1\le n\le N}d_n^2\le C\,N(\log N)^2$ for all large $N$.
Acceptance. FULLY RESOLVES: a complete unconditional proof that $\sum_{n\le N}d_n^2\ll N(\log N)^2$ — machine-checkable (Lean/Coq) preferred, otherwise a full written proof. A proof assuming RH or another unproven hypothesis must flag it explicitly and counts only as an ADVANCES. ADVANCES (each a complete proof, strictly better than the RH-conditional frontier stated in the background — Cramér's $N(\log N)^4$ and Selberg's $\sum d_n^2/n\ll(\log N)^4$): an upper bound $\sum_{n\le N}d_n^2\ll N(\log N)^{\theta}$ with an explicit exponent $\theta<4$ (unconditional, or RH-conditional and improving Cramér's exponent); or an unconditional pointwise gap bound $d_n\ll n^{1/2}(\log n)^{O(1)}$ implying progress toward the sum. Deliver a machine-checkable or complete written proof with explicit exponents.
Background
Posed by Erdős [Er40, p.440], [Er55c, p.2], [Er65b, p.205]; listed as open on erdosproblems.com/233 (fetched 2026-07-21, status 'open'). The prime number theorem gives the matching lower bound $\sum_{n\le N}d_n^2\gg N(\log N)^2$, so the conjecture asserts that this trivial lower bound is the true order of magnitude. The conjectured upper bound would imply $d_n\ll n^{1/2}\log n$ for every $n$, a pointwise gap bound currently known only under the Riemann Hypothesis. Conditional results: Cramér [Cr36] proved $\sum_{n\le N}d_n^2\ll N(\log N)^4$ on RH, and Selberg [Se43] improved this (still on RH) to $\sum_{n\le N}d_n^2/n\ll(\log N)^4$. The partial sums are tabulated in OEIS A074741. The problem is item A8 in Guy's collection [Gu04], and the statement has a Lean formalization in the formal-conjectures project; no Erdős prize is attached. Attacker's tool: analytic number theory (second moments of primes in short intervals, RH-conditional bounds, moment methods in the style of Cramér and Selberg); high-precision computation of $\sum_{n\le N}d_n^2/(N(\log N)^2)$ over extended ranges (A074741) can pin the empirical constant but cannot prove the bound.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Erdős Problem #233 (T. F. Bloom) | website |
| REF-02 | OEIS A074741 — values of the sum of squares of prime gaps | website |
| REF-03 | Lean formalisation (formal-conjectures, Erdős #233) | website |
Investigations · 0
No published investigations yet. This problem is unclaimed territory.