The Binary Totient Series

Will Cook

Long paper

The Binary Totient Series

Erdős #249 By Will Cook July 2026; revised 1 October 2026 Authorship, AI use and citation

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Summary. The dyadic totient sections through level $e \geq 1$ have an explicit rational basis of size $2^{e} + 1$ . Truncating a scaled tail difference produces an integer whose residue certifies nonintegrality when its distances from both endpoints of the residue interval exceed an explicit error bound. The record develops the resulting finite denominator exclusions, equivalent residue criteria, and sufficient exponential-sum estimates. Rational comparison sequences identify the limitations of particular size, parity, rank, and finite-prefix arguments. The short paper supplies the all-base basis theorem; here each conditional result carries the arithmetic hypothesis still needed to apply it. Those hypotheses are not established in the form required to prove irrationality of S=∑ $φ (n) 2^{- n}$ .

This paper covers a claim-bounded problem-specific reasoning surface for Erdős #249: selected checked premises, finite certificates, 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 #249, which remains open.

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In this paper

Square-CRT factorisations

For a prime $p$ and $m \geq 1$ , the identity

φ (p m) = (p - 1) φ (m) + 1_{p ∣ m} φ (m)

separates the coprime and divisible cases. Prescribing a cofactor modulo $p$ by a congruence modulo $p^{2}$ can remove the second term over a chosen finite range. It simplifies the product formula, but does not show that differences between the resulting values are nonzero.

The examples in Observation 6.117 show both possibilities. At $n = 52$ , the offset pairs $(13, 13)$ and $(26, 18)$ give respectively $φ (65) - φ (65) = 0$ and $φ (78) - φ (70) = 0$ (squareCRT_clean_block_can_vanish). All four prime–cofactor products are coprime (squareCRT_vanishing_countermodel_is_clean). At $n = 27$ , the offset pairs $(3, 3)$ and $(6, 8)$ instead give $φ (30) - φ (30) = 0$ and $φ (33) - φ (35) = - 4$ (squareCRT_clean_block_can_be_nonzero), again with coprime factors (squareCRT_nonzero_countermodel_is_clean). These finite equalities are checked by exact arithmetic. No minimality claim about either witness is needed. The general existence theorem exists_squareCRT_clean_horizon supplies a common base for the stated finite congruence requirements, not a nonvanishing assertion.

The vanishing example disproves the implication from correction-free factorisation to nonvanishing; the other example shows that this factorisation does not force vanishing either. This does not establish formal independence of two propositions, or exclude an argument using additional hypotheses. The prime-dilation identity totient_mul_prime_cocycle and the CRT constructions exists_squareCRT_base, exists_large_squareCRT_base remain available with their own prime and residue hypotheses.

A separate elementary identity in the same file concerns alternating sums. Let $I$ be finite, let $i \notin I$ , and suppose an integer-valued function $f$ on finite subsets satisfies $f (T \cup {i}) = f (T)$ for every $T \subseteq I$ . Pairing $T$ with $T \cup {i}$ gives

\sum_{T \subseteq I \cup {i}} (- 1)^{| T |} f (T) = 0.

This identity is proved in Lean (alternating_powerset_sum_eq_zero_of_insert_invariant). It has no totient hypothesis, but an application must still verify invariance under the indicated insertion. Neither it nor the congruence construction supplies the missing nonzero totient difference at a prescribed LCM height.

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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