Reciprocal Mersenne Subseries

Will Cook

Long paper

Reciprocal Mersenne Subseries

Erdős #257 By Will Cook 30 September 2026 Authorship, AI use and citation

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Summary. A weighted summability criterion proves irrationality of reciprocal Mersenne subseries: if h is the part of a supported on a finite nonempty prime set P, every infinite A with ∑ $h (a) / (a (b^{h (a)} - 1))$ finite has irrational ∑ $(b^{a} - 1)^{- 1}$ . The base-two condition gives the conclusion at every integer base for every infinite subset, and permits an example with divergent reciprocal sum, while excluding full support and all odd exponents. The proof averages over a finite part of the support and over dyadic observation lengths. Erdős's weaker reciprocal-summable theorem is proved in full, and a common window extends both arguments to unions of weighted and positive-cover supports. Finite-denominator and achievement-set calculations formulate tests for the targets $1 / 2$ and $1 / 21$ ; those tests do not decide either target or the universal question. Prime support at base 2 and squarefree support at power-of-two bases are attributed as existing work.

This paper covers a claim-bounded problem-specific reasoning surface for Erdős #257: selected checked premises, finite computations, closed routes, surviving obligations, and their exact evidence bands.

It does not establish the validity of claims tagged Lean, which belongs to the cited kernel-checked source, or a solution to Erdős #257, whose universal and half-value questions remain open.

Start here, by the editorial route

In this paper

What survives without arithmetic restriction

Proposition 13.3 (ordinary initial intervals). For every finite nonempty $F$ , integer $X \geq 1$ , and $0 < t \leq 1$ ,

\frac{1}{X} # {1 \leq N \leq X : U_{F} (N) > t} \leq \frac{2}{\log (4 / 3)} κ_{1} (F; t) .

Proof. Choose a positive logarithmic divisor majorant $g (n) = \sum_{d ∣ n} c_{d}$ of cost $K = \sum_{d ∣ Q} c_{d} / d$ , putting $c_{d} = 0$ when $d ∤ Q$ , and let $a = \log (4 / 3)$ . For positive integers $n, Y$ ,

g (n) \leq K n, \sum_{n = 1}^{Y} g (n) = \sum_{d ∣ Q} c_{d} ⌊ Y / d ⌋ \leq K Y .

If $U_{F} (N) > t$ , there is an $r \geq 1$ with $\log (1 + f_{F} (N + r) / t) > a r$ . Otherwise every $f_{F} (N + r) / t \leq (4 / 3)^{r} - 1$ , and

\frac{U_{F} (N)}{t} \leq \sum_{r \geq 1} 2^{- r} ((4 / 3)^{r} - 1) = 1,

a contradiction. If $K \geq a / 2$ the required counting bound is trivial. For $K < a / 2$ , a witnessing integer $n = N + r$ satisfies $a (n - N) < g (n) \leq K n$ , whence $n < a N / (a - K) \leq 2 X$ when $N \leq X$ . A fixed $n$ can witness at most $g (n) / a$ integer values of $N$ , since $1 \leq n - N < g (n) / a$ . The union bound thus gives

# {1 \leq N \leq X : U_{F} (N) > t} \leq \frac{1}{a} \sum_{n = 1}^{2 X} g (n) \leq \frac{2 K X}{a} .

Take the infimum over the admissible majorants. ◻

The proposition also bounds every $L = 1$ dyadic observation window, since such a window averages ordinary initial-interval means. It does not bound conditional arithmetic sampling uniformly in $L$ . Freezing a finite initial support in the irrationality argument requires exactly that arithmetic restriction. An irrationality criterion with logarithmic divisor costs in place of fractional ones therefore does not follow from this proof; it is not disproved either. The corollary above separates finite functionals, not infinite-support irrationality classes. The weighted, fractional-cover, mixed and finite-period noncollapse results are not affected. No conclusion here decides universal #257 or membership of $1 / 2$ or $1 / 21$ .

About this paper

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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