smoothed-novelty

• The novelty meter: the totient sum read through a test function, and the exponent the error carries against the same sum read through a sharp cutoff.
• The object is S_f(y) = sum_n phi(n) f(ny) for f supported on [1, 2], main term (6/pi^2) F(2) y^-2 with F(s) = int f(u) u^(s-1) du, and the meter is E_f(y) = y^2 S_f(y) - (6/pi^2) F(2) read as a power of q = y^2.
• Three test functions: the indicator of [1, 2], the C^2 bump 64 (u-1)^3 (2-u)^3 whose Mellin transform is the closed form 64 sum_k a_k (2^(s+k) - 1)/(s+k), and the C^infinity bump exp(4 - 1/((u-1)(2-u))) whose Mellin transform is a 2000-node Gauss-Legendre quadrature.
• totients sieves phi(n) to 3 * 10^7 in numpy, dividing each prime out of every multiple and finishing with the one prime factor above the square root; checked against brute-force gcd counts to 2000 and against the Mobius route sum_d mu(d) T(floor(x/d)) at x = 10^6, both exact.
• The grid is y = 2^-j for j from 8 to 23.5 in steps of 1/16, 249 samples, the largest window summing the 1.19 * 10^7 scales between 2^23.5 and 2^24.5.
• slopes fits log of the per-octave root mean square of E_f against log q over all 16 octaves and over the lower and upper 8, an octave being the samples with j in [k, k + 1), 16 of them except the last, [23, 23.5], which holds 9, each placed at j = k + 1/2; it prints each slope with its window and residual, and the least-squares slope over every sample beside it.
• zeros_from_pari calls gp -q once for the 138 zeros of zeta to height 300 with zeta(rho - 1) and zeta'(rho) at each, and the explicit formula E_f(y) = 2 Re sum_rho F(rho) zeta(rho - 1)/zeta'(rho) y^(2 - rho) is compared with the measured E_f over the whole grid at 1, 10, 30 and 138 zeros; the least-squares power of |c_rho| against gamma on the 138 zeros is printed for each bump.
• The summation noise is printed at the smallest y as the gap between numpy's pairwise sum and math.fsum of the same products.

WHAT IT PRINTS

• The per-octave slopes in q: indicator 0.5023 (residual 0.179; windows 0.5296 and 0.4883, residuals 0.161 and 0.169), C^2 bump 0.7498 (residual 0.092; windows 0.7553 and 0.7487), C^infinity bump 0.7471 (residual 0.115; windows 0.7474 and 0.7460).
• The explicit formula: with 138 zeros the C^infinity bump's E_f is reproduced to a relative 2.0e-06 over the grid, the C^2 bump's to 2.3e-04, the gap being the tail of its coefficients, whose printed least-squares power against gamma is -3.22 on the 138 zeros; one zero alone gives 0.50 and 0.33.
• |E_f|/y^(3/2) stays between 1.3e-04 and 0.558 for the C^infinity bump against 2 sum |c_rho| = 0.755 over the same 138 zeros.
• The sharp cutoff: the totient error sum_{n <= x} phi(n) - 3x^2/pi^2 jumps by phi(p) - 3(2p - 1)/pi^2 across the prime p, 392073.4 at p = 1000003 and 3920735.7 at p = 10000019.
• Summation noise at j = 23.5: 5.6e-17 against |E_f| = 5.4e-12.

RUN

• uv run python research/lab/py/smoothed-novelty/smoothed_novelty.py
• From the repository root, one core, 15 seconds: 2 for the sieve, 8 for the 498 smoothed sums, under 1 for PARI. Needs gp on the path.
• Nothing is written to disk.

WITNESSES

• stack.md, "The novelty meter": the three slopes with their windows, the explicit formula and its residuals, the jump at a prime.
• The Mellin transform of the C^2 bump at s = 2 is 24/35, printed from the closed form beside the exact value.