Prime-Gap Dyadic Series: Perturbations, Exact Criteria and Certificates

Will Cook

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Prime-Gap Dyadic Series: Perturbations, Exact Criteria and Certificates

Erdős #251 By Will Cook September 2026 Authorship, AI use and citation

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Summary. A convergent dyadic series with nonnegative integer coefficients can be made rational by sparse nonnegative corrections of upper Banach density zero, eventually bounded by any f(n)→∞, while preserving every fixed modulus and, for a logarithmic allowance, short-block empirical distributions. For prime gaps the cumulative positions remain asymptotic to n log n; no primality conclusion is asserted. Summation by parts reduces the prime series to the gap series, and the tails $T_{N} = \sum_{j \geq 1} g_{N + j} 2^{- j}$ satisfy $T_{N + 1} = 2 T_{N} - g_{N + 1}$ . That recurrence characterises rationality and yields finite tests with explicit remainder bounds. An exact finite certificate shows that a rational value of the prime series would have denominator at least $2^{39998} > 10^{12040}$ . The bound does not establish irrationality.

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Numerical counts and proportions

The scan in Section 9 used the $6 841 648$ primes below $1.2 \times 10^{8}$ and computed double-precision tails for offsets $h \leq 16$ . We reproduce its reported counts for $h = 1, \dots, 4$ . The denominator of each proportion is the number of tested basepoints, $6 841 565 - h$ , rather than the prime count.

$h$	events	proportion	$δ = + 2$	$δ = - 2$	last event at prime
1	56 427	0.008248	28 022	28 405	119 995 753
2	31 979	0.004674	15 742	16 237	119 993 807
3	30 233	0.004419	15 093	15 140	119 998 321
4	29 262	0.004277	14 606	14 656	119 993 473

These are separate counts for each offset $1 \leq h \leq 16$ , not counts of indices successful for all offsets at once. The proportions range from $0.00418$ to $0.008248$ , with nearly balanced signs at each offset. For $h = 1$ , the first row below gives the proportions in the eight bands; the second multiplies each by its band’s mean of $\log p$ :

\begin{array}{l} 0.011285, 0.008951, 0.008303, 0.007936, 0.007651, 0.007507, 0.007253, 0.007094; \\ 0.17671, 0.15058, 0.14420, 0.14067, 0.13766, 0.13667, 0.13333, 0.13148 . \end{array}

The ratio of the last rescaled proportion to the first is $0.744$ at $h = 1$ and lies between $0.744$ and $0.8121$ across the measured offsets. Theorem 8.7 gives limiting density zero for each fixed offset. The bound $O_{h} (X / \log \log X)$ derived after its proof is only an upper bound; it predicts neither a monotone decline nor an asymptotic proportion. The finite-band ratios therefore do not establish an asymptotic rate.

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Authorship and AI use. Will Cook built and directed the research infrastructure and maintains the public release. He reviewed claims when he could. AI agents did most of the research and drafting. Cook did not independently verify every claim.

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