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.

Start here, by the editorial route

In this paper

The full-block exponential-sum estimate

The required inequality.

\forall h \geq 1 \forall X_{0} \exists X, L : max (X_{0}, 1) \leq X, 16 (2 X + h + L + 2) \leq 2^{L}, ‖ \sum_{N = X}^{2 X - 1} e (D (h, N, L) / 2^{L}) ‖ \leq \frac{21}{25} X,

where $D (h, N, L) = \sum_{j < L} (φ (N + h + 1 + j) - φ (N + 1 + j)) 2^{L - 1 - j}$ . This inequality is unproved; it implies irrationality (Erdos249257.TotientTailPeriodKiller.DTWFirstHarmonicNormGap, Erdos249257.irrational_totient_series_of_first_harmonic_norm_gap). By Proposition 12.2 the real-part bound with constant $89 / 100$ for the exact doubling phases suffices; its transfer to the truncated phases uses the separate error estimate in that proposition.

The discrepancy in terms of the doubling map.

For $L \geq h$ , dividing by $2^{L}$ and reindexing the two ranges gives

\begin{matrix} (3) & \frac{D (h, N, L)}{2^{L}} = \underset{head}{\underset{⏟}{- \sum_{t = 1}^{h} \frac{φ (N + t)}{2^{t}}}} + \underset{body}{\underset{⏟}{(2^{h} - 1) \sum_{t = h + 1}^{L} \frac{φ (N + t)}{2^{t}}}} + \underset{tail}{\underset{⏟}{\sum_{t = L + 1}^{L + h} \frac{φ (N + t)}{2^{t - h}}}} . \end{matrix}

The phase is a geometrically weighted sum of totients at $L + h$ consecutive arguments. In the first two ranges, $1 \leq t \leq L$ , the coefficient has denominator $2^{t}$ and odd numerator. Its contribution is therefore integral exactly when $v_{2} (φ (N + t)) \geq t$ . In the last range, $L < t \leq L + h$ , the denominator is instead $2^{t - h}$ , so the corresponding threshold is $t - h$ . Here $v_{2} (a)$ denotes the exponent of $2$ in the positive integer $a$ . The selected offset in the four-term decomposition lies in the middle range, where the odd multiplier is $2^{h} - 1$ .

Relevant comparison estimates.

The following comparisons identify the estimates still needed; they do not rule out these methods.

Weyl differencing and van der Corput. The van der Corput inequality applies to arbitrary bounded complex sequences [23], but its use here requires estimates for shifted correlations. For the exact phases $x_{N} = 2^{N} α_{h}$ of Lemma 12.1,

x_{N + q} - x_{N} = (2^{q} - 1) x_{N} .

Thus differencing gives another doubling orbit, not a polynomial phase of lower degree. The finite phase $D (h, N, L) / 2^{L}$ approximates this orbit modulo $1$ ; it is not the exact orbit itself. No bound for the resulting correlations is established here.

The large sieve. A proposed application would need a suitable family of separated frequencies and a way to recover a bound for each fixed $h$ . Averaging only over $h$ would not suffice: the condition asks for a block estimate for every $h$ , not for most of them.

Bilinear sums over prime factors. Factoring $N + t = m p$ and using $φ (m p) = φ (m) (p - 1)$ makes the selected totient contribution linear in the prime $p$ . Section 12.3 gives this decomposition and identifies the remaining weight. A bound for that weight is still needed before a Vaughan- or Vinogradov-type argument can be applied.

Erdős’s 1948 digit method. The comparison series is $E = \sum_{n \geq 1} 1 / (2^{n} - 1) = \sum_{N \geq 1} d (N) / 2^{N}$ , whose irrationality Erdős proved [11]. For the totient series, the corresponding Möbius expansion is exact: $φ = μ * id$ gives

\begin{matrix} (4) & S = \sum_{d \geq 1} μ (d) \frac{2^{d}}{(2^{d} - 1)^{2}} \end{matrix}

The identity follows by Möbius inversion; a numerical computation gives $S = 1.3676308019850223 \dots$ to sixteen digits.

Erdős combines congruences forcing increasing powers of $2$ to divide a block of divisor counts with an averaged tail bound, producing long zero blocks in the binary expansion [11]. To adapt that argument here, one would need congruences for the actual totient coefficients together with a compatible tail estimate. The identities alone provide neither combination. In (4), the coefficients $μ (d)$ can have either sign, so positivity of the original totient series does not give positivity term by term in this expansion. This is a limitation of the proposed direct transfer, not a proof that a digit argument or a different use of the Möbius expansion cannot work.

The unresolved estimate.

For general coefficients $0 \leq c (n) \leq n$ , irrationality alone does not imply the block norm condition. The following example has very sparse nonzero coefficients. It is irrational, but almost all phases in each sufficiently large block lie close to $1$ .

Theorem 12.5 (The block norm condition is stronger in the class $0 \leq c (n) \leq n$ ). Let $c (n) = 1$ if $n = k!$ for some $k \geq 1$ and $c (n) = 0$ otherwise, so $0 \leq c (n) \leq n$ for all $n \geq 1$ , and let $β = \sum_{n \geq 1} c (n) / 2^{n} = \sum_{k \geq 1} 2^{- k!}$ . Then $β$ is irrational, and for every $h \geq 1$ and every $X \geq 81 (h + 5)$ ,

\sum_{N = X}^{2 X - 1} \cos (2 π 2^{N} (2^{h} - 1) β) > \frac{9}{10} X .

Consequently the block norm condition fails for $β$ at every sufficiently large scale, for every admissible truncation depth, even though $β$ is irrational. Thus irrationality, nonnegative integer coefficients and the bound $c (n) \leq n$ do not imply the block gap. A proof for $S$ needs an additional property not shared by this example; the comparison does not specify which additional property will suffice.

Proof. The binary expansion of $β$ has infinitely many ones separated by unbounded strings of zeros. It is not eventually periodic, so $β$ is irrational. Write $γ = (2^{h} - 1) β$ , and let $k!$ be the least factorial greater than $N$ . The earlier terms become integers after multiplication by $2^{N}$ , so

2^{N} γ \equiv (2^{h} - 1) \sum_{j \geq k} 2^{N - j!} (\mod Z) .

If $N \leq k! - h - 5$ , the expression on the right lies between $0$ and $2 \cdot 2^{N + h - k!} \leq 1 / 16$ . Thus $‖ 2^{N} γ ‖_{R / Z} \leq 1 / 16$ and $\cos (2 π 2^{N} γ) \geq \cos (π / 8) > 9238 / 10000$ . The remaining $N$ lie in $(k! - h - 5, k!]$ , at most $h + 5$ values per factorial. The relevant factorials have ratio at least $3$ , and $h + 5 < X$ , so $[X, 2 X)$ meets at most one such interval. Hence the sum is greater than

\frac{9238}{10000} (X - h - 5) - (h + 5) \geq \frac{9}{10} X + \frac{h + 5}{250}

when $X \geq 81 (h + 5)$ .

To pass from these exact phases to the finite residue test, note that $0 \leq c (n) \leq 1$ puts each scaled tail in $[0, 1]$ . The error in the truncated tail difference is therefore at most $2^{- L}$ , so each complex phase changes by at most $2 π 2^{- L}$ . At an admissible depth, $2^{L} \geq 16 (2 X + h + L + 2) > 32 X$ , and the total change in the block sum is less than $π / 16$ . Its real part is consequently greater than $9 X / 10 - π / 16 > 21 X / 25$ for $X \geq 81 (h + 5)$ . Its norm cannot satisfy the $21 X / 25$ bound either. ◻

This example rules out the implication from irrationality and the stated coefficient bound to the block norm condition. It does not say that the totient sum is Liouville-like, or that every argument based on its digits fails. A proof of the block estimate must use information that distinguishes the actual totient phases from this comparison sequence.

The size of the required saving.

Under Lebesgue measure on the phase circle, the binary digit maps are independent fair bits. The digit-change proportion therefore tends to $1 / 2$ for almost every phase. For almost every phase, the real-part block sum is also $O (\sqrt{X \log \log X})$ , by the law of the iterated logarithm of Erdős and Gál for lacunary series [29]. The block bound follows by subtracting two initial partial sums. An almost-everywhere theorem supplies no conclusion at the particular phase $α_{h} = (2^{h} - 1) S$ . The requirement is $ρ_{h} \geq 0.11$ and a saving of a constant factor. The finite values in Table 1 have $ρ_{h}$ near $0.50$ and small block averages. They are compatible with square-root cancellation, but do not establish its asymptotic order. The sufficient condition asks only for the stated fixed saving on arbitrarily large blocks. No estimate in this record supplies it for the totient phases.

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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