Computation of the Möbius-Prime Sum

Title: Computation of the Möbius-Prime Sum ∑₍≤₍ μ (n) pₙ to N = 10¹¹ Authors: Muhamad Wakid (ORCID: 0009-0008-6274-778X) Upload type: Preprint Description: This record contains the manuscript, source code, data, and figures for a computational study of the prime-indexed Möbius sum S (N) = ∑₍≤₍ μ (n) pₙ, where μ is the Möbius function and pₙ is the n-th prime. The sum is computed exactly for all N ≤ 10¹¹: every index is scanned through 10⁸, and the range 10⁸ < N ≤ 10¹¹ is covered by verified segmented blocks of length 2×10⁷. Unlike the Mertens function M (x), for which sublinear methods reach far larger arguments, S (N) is a functional of the entire Mertens path and requires full enumeration. The main result is structural. Writing each prime gap as its mean plus a fluctuation, g₍+₁ = log pₙ + eₙ, Abel summation gives the exact identity S (N) = A (N) − B (N), where A (N) = pN M (N) − ∑ M (n) log pₙ is a smoothly weighted functional of the Mertens path and B (N) = ∑ M (n) eₙ carries all gap-specific information. Computed exactly to 10⁹, the normalized fluctuation |B (N) |/√EN (with EN = ∑ μ (n) ²pₙ²) remains below 1. 5×10⁻⁵ at the checkpoints from 10⁷ to 10⁹ and equals 6. 2×10⁻⁷ at 10⁹, decaying consistently with N^−1/2. On the self-normalized scale the prime-gap fluctuations therefore contribute only in the fifth-to-seventh decimal place: to the computed resolution, S (N) is governed almost entirely by the smooth Mertens component, and the several gap–Mertens correlation diagnostics measured are consistent with zero for this reason. This is a negative result about the arithmetic content of S (N), stated as such. At N = 10¹¹ the verified terminal values are pN = 2, 760, 727, 302, 517, M (N) = −87, 856, S (N) = −232, 607, 877, 294, 603, 985, and RE (N) = S (N) /√EN = −0. 5993596357. The computation passes through the independently published values M (10⁸), M (10⁹), M (10¹⁰) and p₁₀℉℉, which confirm both sieve inputs at the terminal scales. All terminal outputs carry SHA-256 checksums. The package is fully reproducible. The exact A/B split is regenerable from source to 10⁹; the full 10¹¹ trajectory is regenerated by the included segmented generator, which verifies the exact prime-gap/Abel identity at every block. Contents: the manuscript (LaTeX + PDF), all C++ and Python sources including the 10¹¹ generator, the retained trajectory and diagnostic CSVs, figures, checksum manifests, and a provenance trail from an earlier draft. Keywords: Möbius function; Mertens function; prime-indexed sums; prime gaps; Abel summation; experimental mathematics; number theory