Computation of the Möbius-Prime Sum

Title: Computation of the Möbius-Prime Sum ∑n≤N μ(n)p_n 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) = ∑n≤N μ(n)p_n, where μ is the Möbius function and p_n 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, gₙ₊₁ = log p_n + e_n, Abel summation gives the exact identity S(N) = A(N) − B(N), where A(N) = p_N M(N) − ∑ M(n) log p_n is a smoothly weighted functional of the Mertens path and B(N) = ∑ M(n) e_n carries all gap-specific information. Computed exactly to 10⁹, the normalized fluctuation |B(N)|/√E_N (with E_N = ∑ μ(n)²p_n²) 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 p_N = 2,760,727,302,517, M(N) = −87,856, S(N) = −232,607,877,294,603,985, and R_E(N) = S(N)/√E_N = −0.5993596357. The computation passes through the independently published values M(10⁸), M(10⁹), M(10¹⁰) and p10¹¹, 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.