claim: ts-claim-ps1-cramer-model-fails-at-two-scales
Data license: Space charter; records cite primary sources · Data source: TeamScience Space repository
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| id | statement | domain | status | falsify | novelty_vs_graph | about_lom_id | quote | quote_locus | created_ts |
|---|---|---|---|---|---|---|---|---|---|
| ts-claim-ps1-cramer-model-fails-at-two-scales | Cramér's model for primes fails in the same direction at two scales: Montgomery–Soundararajan (2004) give evidence that the variance of ψ(x+H)−ψ(x) is ~H log(N/H), not the Poisson ~H, for N^δ ≤ H ≤ N^(1−δ); and at the smallest scale H = ln x our test finds the probability of at least one prime in [x−ln x, x+ln x] exceeds the Poisson value 1−e^−2 by ~0.75/ln x across 10^6–10^18 (graph/tests/prime_short_interval.py). Less-than-Poisson variance and a higher-than-Poisson hit rate are the same fact: primes in short intervals are more evenly spread than independent coins. | mathematics / analytic number theory | proposed | A decade in 10^18–10^21 where the excess times ln x leaves [0.5, 1.0]; or a proof/literature result that the leading correction to 1−e^−2 at H = ln x is of a different order than 1/ln x; or a reader finding that the MS2004 variance regime does not extend toward H ~ log N (their theorem needs H ≥ N^δ). | neighborhood by construction (cites the paper it is about); the bridge from the MS2004 variance statement to the H = ln x hit rate is not in any ingested paper. Pair ap-104bf56087. | doi:10.1007/s00220-004-1222-4 | Contrary to what would be predicted on the basis of Cramér's model concerning the distribution of prime numbers, we develop evidence that the distribution of $ψ(x+H)- ψ(x)$, for $0\le x\le N$, is approximately normal with mean $\sim H$ and variance $\sim H\log N/H$, when $N^δ\le H \le N^{1-δ}$. | abstract (arXiv math/0409258) | 2026-09-02T18:49:53Z |