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

Column rescaling and cancellation

A nonzero determinant need not prevent one row from being constant. For example,

(\begin{matrix} 1 & 1 \\ 1 & - 1 \end{matrix}) (\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}) = (\begin{matrix} 1 & - 1 \\ 1 & 1 \end{matrix}) .

The first and last matrices have determinants $- 2$ and $2$ , respectively; the second row of the resulting matrix is constant. This simple column rescaling matters when a determinant is proposed as evidence for cancellation in the first-harmonic sums.

Proposition 10.4 (A nonzero minor survives inverse-phase column weights). Let $d \geq 1$ be an integer, let $e_{0}, \dots, e_{d - 1}$ be nonnegative integers, and let $z_{0}, \dots, z_{d - 1}$ be nonzero complex numbers. Write $P_{i j} = z_{j}^{e_{i}}$ , and suppose $e_{i_{0}} = 1$ for some $i_{0}$ . Multiplying column $j$ by $W_{j} = z_{j}^{- 1}$ gives

(P diag (W))_{i_{0} j} = 1, det (P diag (W)) = det (P) \prod_{j = 0}^{d - 1} z_{j}^{- 1} .

Thus a nonzero determinant remains nonzero while row $i_{0}$ becomes constant. If all $| z_{j} | = 1$ , its absolute value is also unchanged.

The entry formula and the determinant of a product prove the statement. If row-dependent weights are allowed, the choice $W_{i j} = z_{j}^{i} / z_{j}^{e_{i}}$ gives $W_{i j} P_{i j} = z_{j}^{i}$ and makes row zero constant. It produces the ordinary Vandermonde matrix, which is nonsingular when the $z_{j}$ are pairwise distinct (rowResidualMonomialMatrix_reconstructs_phasePowerMatrix, rowReconstruction_zero_row_one).

These are statements about complex matrices, not arithmetic properties of the totient sequence. In particular, determinant nonvanishing, even with unit-modulus weights, does not by itself exclude complete alignment in the indicated row. An application of the first-harmonic criterion (irrational_totient_series_of_first_harmonic_norm_gap) needs additional information that excludes such weights or controls the resulting weighted sum. The example does not require that information to take the form of an identity coupling different rows, and it does not rule out determinant arguments that use appropriate additional hypotheses.

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.

Cite and contact. Cite this paper by its title, author and date above, with its PDF; cite the earlier sources it uses for a mathematical result. For software, use the release citation and give the commit used. Contact Will with questions or corrections.

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