shifted-sums

• Walks the shifted sums T_s = sum_{w in B_s} mu(6w + 1) of the Franel note's repunit section, B_s the 2^s integers below 3^s whose base 3 digits lie in {0,1}, to s = 26.
• The walk takes w in ascending order, so T_s is the prefix sum P_n at n = 2^s; it records every sign change of the last nonzero sign of P_n for n <= 2^26.
• The control is the whole class: M(x; 6, 1) = sum_{n <= x, n = 1 mod 6} mu(n) at x = 3^(s+1) - 2, the top of the image at level s, to 3^16 - 2, beside the drift 3/4 - Psi_(2,3)(x), Psi_(2,3) the count of 3-smooth numbers up to x: the term at s = 0 of its Dirichlet series (1/2)(1/(zeta(s)(1 - 2^-s)(1 - 3^-s)) + 1/(L(s, chi_-3)(1 + 2^-s))) when each M(x/(2^a 3^b)) is read as its constant -2 and 1/(L(0, chi_-3)(1 + 2^0)) = 3/2.

METHOD

• PARI moebius, batched from Python into 8 gp -q processes; the walk is cut into chunks of 2^16 consecutive indices, chunk j holding w = w(j) 3^16 + w(r) with w(i) the base 3 reading of the binary digits of i.
• Pass one returns each chunk's sum and the minimum and maximum of its local prefix; a chunk can hold a sign change only if its prefix range reaches the side opposite the last nonzero sign, and pass two rewalks exactly those chunks from their true offsets.
• Self-checks, each from an independent source: T_1 to T_16 against the list printed on the note; T_(t-1) - T_(t-2) + mu(3^t + 1) against the note's endpoint readings M_F(x_t; R_t) at t = 2 to 24, which restricted-franel walks over a different set; a plain sequential walk to 2^18 repeats T_18, the flip count and the last flip.

RUN

• uv run python research/lab/py/shifted-sums/shifted_sums.py from the repository root; append walk or control for one verb.
• walk 83 s on 8 cores (449 s of CPU), control 7.7 s on one; gp on the path.

WITNESSES

• The Franel note, the repunits: T_s at s = 17 to 26 reading -88, 121, 665, 1523, 1857, 1450, 4479, 4939, 1417, 12641, negative at s = 2 to 4 and 6 to 17, zero at s = 1, 5, positive at s = 18 to 26, abs T_s / 2^(s/2) largest at 1.8562 at s = 13.
• The prefix walk: 132 sign changes to n = 2^26, the last at n = 238418, element 6w + 1 = 1128943015, never negative from there to the cut and 0 at n = 238419.
• The control: M(x; 6, 1) negative at s = 2 to 13, 8 at s = 14, 265 at s = 15, 3996 sign changes below 3^16; the drift reads -173.25 at s = 13 against -169 and -223.25 at s = 15 against 265.