The Binary Totient Series

Relations between the representations

Binary tails, Möbius sums, rational approximations and exponential sums express related information about $S$. We give the identities that connect them and then check which assumptions survive each change. This is necessary before transferring a counterexample to another representation, as the finite-state decoding and polynomial denominator examples illustrate.

Binary-weighted series and tails.

The coefficients in $S=\sum _{n\ge 1}\phi (n)2^{-n}$ are unbounded; this expression is not the base-two digit expansion of $S$. The tail $R_N$, finite difference numerator $D(h,N,L)$ and test $\mathcal{C}(h,N,L)$ retain this coefficient information. The identity

$$S=\frac{1}{2}+\sum _{d\ge 1}\frac{\mu (d)}{(2^d-1{)}^2}$$

(tsum_totient_half_pow_eq_half_add_moebius_sq) and its partial-sum version (mobius_square_series_eq_partial_add_tail) express the same value by Möbius sums. An equality of values does not by itself transfer a claim about the original coefficient subsequences.

Lambert sums and finite numerator polynomials.

The Lambert identities for $\mu$, $1$ and $\phi$ give the series relations in Section 5. For a finite divisor sum, the polynomial evaluation (mobiusNumeratorPolynomial_eval_two) and the definition (mobiusNumerator) put its numerator into integer polynomial arithmetic. This evaluates a finite contribution, not the complementary infinite tail.

Coprimality probability.

Let $X,Y$ be independent with $Pr(X=n)=Pr(Y=n)=2^{-n}$ for $n\ge 1$. Then $S-\frac{1}{2}=Pr(gcd(X,Y)=1)$. The divisibility probabilities are $Pr(d\mid X,d\mid Y)=(2^d-1{)}^{-2}$ (tsum_pos_pair_both_dvd_half_eq_inv_mersenne_sq). Möbius inversion converts these to the coprimality probability. The decomposition by the greatest common divisor also gives the weighted coprime-pair identity (tsum_pos_coprime_inv_mersenne_eq_one). Counting coprime pairs by their sum gives the totient expression directly (tsum_coprime_pair_pow_eq_tsum_totient_mul_pow).

Cyclotomic factors of finite denominators.

The coefficient formula for $P_r$ (mobiusNumeratorPolynomial_coeff) and the cyclotomic divisibility statement (cyclotomicValue_dvd_baseMobiusShadow_den) identify factors in the reduced denominators of the specified finite sums. The product-divisibility theorem (upperHalfChannel_product_dvd_den_of_scale_primeFactors_le) is used in the exact denominator formula (lcmHeight_scaledMobiusShadow_den_exact). Growing finite denominators alone do not exclude cancellation with the remaining tail.

Numerator divisibility and denominator bounds.

Write $x=a/b$ in lowest terms, with $b>0$. For $c\in \mathbb{Z}$ and a positive divisor $H$ of $b$, if the reduced denominator of $cx$ divides $H$, then $b/H\mid c$ (scalarLocalization_complement_dvd). In particular, $|c|\ge b/H$ when $c\ne 0$. A related bound is

$$0<\frac{a}{b}<\frac{2}{2^n-1},2^r\mid a,n\ge 1⟹2^r(2^n-1)<2b$$

for integers $r\ge 0$ (positiveRat_mersenne_height). Here positivity gives $a\ge 2^r$; the size inequality then gives the conclusion. These are arithmetic bounds on a reduced fraction, not irrationality criteria without an additional approximation estimate.

Farey intervals.

The denominator bound (farey_gap) applied to the explicit window $(N,K)=(1,240)$ gives (gap_check_window_1_240_le_79639646646701375323355774875831053) and hence $S\ne p/q$ for integers $p$ and $1\le q\le Q_0$. The finite input is a totient numerator modulo $2^{240}$. Its exact first failing denominator is $Q_0+1$, the denominator of the relevant Farey mediant (gap_check_window_1_240_first_failure). This last assertion concerns this fixed window.

Differences at least-common-multiple shifts.

For $H_t=\mathrm{lcm}(1,\dots ,t)$ (periodLcm), a jump in $H_t$ can occur only at a prime power (eq_prime_pow_of_not_dvd_periodLcm). The identities (lcmRayArithmeticLetter_eq_deltaTotient) and (diagonalWindowIncrement_eq_lcmRayArithmeticLetter) identify the coefficient difference as ${\delta }_t(j)=\phi (2H_t+j)-\phi (H_t+j)$ and give $D(H_t,H_t,L)=\sum _{j=1}^L{\delta }_t(j)2^{L-j}$. Thus these arithmetic formulas concern particular instances of the same finite residue test; additional window restrictions remain in force.

Exponential sums of the window residues.

For integers $h,X\ge 1$, the relevant block statistic is

$$\sum _{N=X}^{2X-1}\cos \left(2\pi \frac{D(h,N,L)}{2^L}\right).$$

A bound by $9X/10$, together with $L\ge h$ and $16(2X+h+L+2)\le 2^L$, yields a certificate in the block (exists_certifiedKill_of_first_harmonic_gap). The linked irrationality implication uses the stronger complex norm bound $21X/25$, for every $h$ on arbitrarily late blocks satisfying the same size inequality (irrational_totient_series_of_first_harmonic_norm_gap). The finite implication uses $\cos (\pi /8)>9/10$; the required cancellation bound is still unproved. The norm bound on the complex exponential sum is sufficient for the displayed real-part bound, but is stronger.