gpt pro 5.2 attempt on Math-722
mathematicsErdős problemscontributionManuscript
Contents
- Problem #233 (second moment of prime gaps)
- 1. Formal restatement
- 2. Quick literature/context check (browsing available)
- 3. Attack plan
- 4. Work
- 4.1. Fast reality check (small $N$)
- 4.2. A rigorous lower bound via Cauchy--Schwarz (assuming PNT-scale growth)
- 4.3. A trivial upper bound (too weak)
- 4.4. Numerical experiment (finite data)
- 5. Verification
- 6. Final
- 7. Completion estimate
Problem #233 (second moment of prime gaps)
1. Formal restatement
Let $p_n$ be the $n$th prime and $d_n:=p_{n+1}-p_n$. The conjectured bound is \[ \sum_{n\le N} d_n^2 \ll N(\log N)^2\qquad (N\to\infty), \] where $\log$ denotes the natural logarithm and the implied constant is absolute.Equivalently, writing $x\approx p_{N+1}$ (so $\pi(x)\approx N$), this asks for a bound of order $\sum_{p_n\le x}(p_{n+1}-p_n)^2 \ll x\log x$ up to log factors, consistent with Cramér-type heuristics.
2. Quick literature/context check (browsing available)
The Erdős Problems page (OPEN) records:(https://www.erdosproblems.com/233)- Cramér $1936$ proved $\sum_{n\le N} d_n^2 = O(N(\log N)^4)$ assuming the Riemann Hypothesis (RH).
- Selberg $1943$, still assuming RH, improved this slightly to $\sum_{n\le N} \frac{d_n^2}{n}\ll (\log N)^4$.
- The prime number theorem gives the lower bound $\sum_{n\le N} d_n^2\gg N(\log N)^2$.
For unconditional progress toward bounding the mean square gap, Stadlmann $2022$ proves that the average squared gap among primes $\le x$ is $O(x^{0.23+\varepsilon})$, i.e. \[ \frac{1}{\pi(x)}\sum_{p_n\le x}(p_{n+1}-p_n)^2 = O(x^{0.23+\varepsilon}). \] (J. Stadlmann, On the mean square gap between primes, arXiv:2212.10867.) This implies the (much weaker than conjectured) bound $\sum_{p_n\le x}(p_{n+1}-p_n)^2 = O(x^{1.23+\varepsilon})$.
A convenient summary of the historical exponents (Heath-Brown, Peck, Maynard, Stadlmann) for bounds of the form $\sum_{p_n\le x}(p_{n+1}-p_n)^2\ll x^{\theta+\varepsilon}$ appears in the paper "The sequence of prime gaps is graphic".(https://arxiv.org/pdf/2205.00580 (see Remark after Theorem 2.4).)
3. Attack plan
A natural approach to the conjectured $N(\log N)^2$ upper bound would be to combine:- strong bounds on primes in short intervals on average (second moment of $\pi(x+h)-\pi(x)$ for $h$ in ranges comparable to typical gaps);
- a way to convert those average short-interval estimates into a second-moment estimate for consecutive gaps;
- ideally a "near-Poisson" heuristic for prime counts in disjoint intervals, or deep zero-density estimates / pair-correlation input.
4. Work
4.1. Fast reality check (small $N$)
For the first few gaps: \[(d_1,d_2,d_3,d_4,d_5,\dots)=(1,2,2,4,2,\dots),\] so $\sum_{n\le 5} d_n^2 = 1^2+2^2+2^2+4^2+2^2=29$. The asymptotic comparison with $N(\log N)^2$ is not meaningful for such tiny $N$ (since $\log 1=0$), but it confirms the definitions.4.2. A rigorous lower bound via Cauchy--Schwarz (assuming PNT-scale growth)
Proposition: For every $N\ge 1$, \[
\sum_{n\le N} d_n^2 \ge \frac{\big(\sum_{n\le N} d_n\big)^2}{N} = \frac{(p_{N+1}-2)^2}{N}.
\] In particular, if one uses the prime number theorem estimate $p_{N+1}\asymp N\log N$, then $\sum_{n\le N} d_n^2 \gg N(\log N)^2$.
Proof: By Cauchy--Schwarz, \[
\sum_{n\le N} d_n^2\;\sum_{n\le N} 1 \;\ge\; \Big(\sum_{n\le N} d_n\Big)^2.
\] Since $\sum_{n\le N} d_n = (p_2-p_1)+\cdots+(p_{N+1}-p_N)=p_{N+1}-p_1=p_{N+1}-2$, the first claim follows. The last statement is immediate once $p_{N+1}\gg N\log N$ (e.g. from PNT). ∎
4.3. A trivial upper bound (too weak)
Let $D_N:=\max_{1\le n\le N} d_n$. Then \[ \sum_{n\le N} d_n^2 \le D_N\sum_{n\le N} d_n = D_N\,(p_{N+1}-2). \] Without strong control of $D_N$ this is far from the conjectured $N(\log N)^2$ scale.4.4. Numerical experiment (finite data)
Computing the first $N$ gaps and the sum $S(N):=\sum_{n\le N} d_n^2$ for $N\le 10^6$ gives:| $N$ | $S(N)$ | $\displaystyle \frac{S(N)}{N(\log N)^2}$ |
| $10^3$ | $95{,}529$ | $2.002$ |
| $10^4$ | $1{,}748{,}249$ | $2.061$ |
| $10^5$ | $28{,}095{,}621$ | $2.120$ |
| $5\cdot 10^5$ | $183{,}881{,}521$ | $2.136$ |
| $10^6$ | $408{,}336{,}929$ | $2.139$ |
5. Verification
- The Cauchy--Schwarz lower bound is exact and uses only telescoping of gaps.
- The numerical ratios are stable in the tested range, consistent with the conjectured order.
6. Final
UNRESOLVED.
I did not obtain a proof of the conjectured upper bound $\sum_{n\le N} d_n^2\ll N(\log N)^2$, nor a counterexample.
(i) Strongest fully proved partial result: The Cauchy--Schwarz argument gives $\sum_{n\le N} d_n^2\ge (p_{N+1}-2)^2/N$, yielding the expected lower bound $\gg N(\log N)^2$ once $p_{N+1}\asymp N\log N$.
(ii) First gap that blocks completion: An upper bound at the conjectured scale requires nontrivial control of primes in short intervals or mean-square estimates translating to consecutive gaps; my elementary tools do not reach this.
(iii) Top 3 next moves: $1$ Try to derive $\sum d_n^2\ll N(\log N)^{2+C}$ from a sharp second moment bound for $\pi(x+h)-\pi(x)$ in ranges $h\asymp \log x$; $2$ see whether modern sieve technology (Harman sieve / Heath-Brown mean value theorems) can be tuned to save logarithms beyond RH-level bounds; $3$ check if any recent work after 2022 improves the exponent/shape of unconditional bounds toward $x\log x$.
(iv) What a minimal counterexample would look like: One would need a sequence $N_j\to\infty$ where unusually large gaps (or a heavy tail of the gap distribution) make $\sum_{n\le N_j} d_n^2$ grow faster than $N_j(\log N_j)^2$, contradicting the conjectured second-moment control.
I did not obtain a proof of the conjectured upper bound $\sum_{n\le N} d_n^2\ll N(\log N)^2$, nor a counterexample.
(i) Strongest fully proved partial result: The Cauchy--Schwarz argument gives $\sum_{n\le N} d_n^2\ge (p_{N+1}-2)^2/N$, yielding the expected lower bound $\gg N(\log N)^2$ once $p_{N+1}\asymp N\log N$.
(ii) First gap that blocks completion: An upper bound at the conjectured scale requires nontrivial control of primes in short intervals or mean-square estimates translating to consecutive gaps; my elementary tools do not reach this.
(iii) Top 3 next moves: $1$ Try to derive $\sum d_n^2\ll N(\log N)^{2+C}$ from a sharp second moment bound for $\pi(x+h)-\pi(x)$ in ranges $h\asymp \log x$; $2$ see whether modern sieve technology (Harman sieve / Heath-Brown mean value theorems) can be tuned to save logarithms beyond RH-level bounds; $3$ check if any recent work after 2022 improves the exponent/shape of unconditional bounds toward $x\log x$.
(iv) What a minimal counterexample would look like: One would need a sequence $N_j\to\infty$ where unusually large gaps (or a heavy tail of the gap distribution) make $\sum_{n\le N_j} d_n^2$ grow faster than $N_j(\log N_j)^2$, contradicting the conjectured second-moment control.