Reconstructing the Prime Zeta Function from Semiprimes: A Stable Nested Radical and Its Condition Number.

Najeem Ziauddin

doi:10.36948/ijfmr.2026.v08i04.85021

Reconstructing the Prime Zeta Function from Semiprimes: A Stable Nested Radical and Its Condition Number.

Author(s)	Mr. Najeem Ziauddin
Country	India
Abstract	Let P(s) denote the prime zeta function and let S₂(s) denote the Dirichlet series over semiprimes, with prime squares included. Separating the ordered products pq according to whether p ≠ q or p = q gives the classical identity 2S₂(s) = P(s)² + P(2s) for Re(s) > 1. The present paper studies its inverse use. Solving recursively for P(s) produces a nested radical in S₂(s), S₂(2s), S₂(4s), and so on, and we ask how much of P(s) is recoverable from a finite initial segment of the semiprime sequence. Our main results concern the numerical stability of this inverse map. We prove that the error of the finite-data reconstruction is bounded by 2T(s)/P(s), where T(s) is the omitted semiprime tail, independently of the recursion depth, and we prove a sharp asymptotic showing that the error equals T(s)/P(s) to leading order, so that the inverse map has condition number exactly 1/P(s) with respect to the tail. Numerically, using the first one million semiprimes, which terminate at 5109839, and 60-digit arithmetic, the measured ratio agrees with 1/P(s) to seven significant digits simultaneously at s = 1.5, 2 and 3. At s = 3 the reconstruction error is 2.0164974 × 10⁻¹⁴. We also record an analytic obstruction: because the recursion inverts a square, its complex-analytic continuation has branch points at the zeros of P, and we locate one at s = 1.44721093994742873 + 38.1082121075686297i, inside Re(s) > 1. Consequently the nested radical cannot be inserted naively into a Perron integral for the prime counting function.
Keywords	Prime Zeta Function; Semiprimes; Almost Primes; Dirichlet Series; Nested Radical; Numerical Stability; Condition Number
Field	Mathematics > Maths + Physics
Published In	Volume 8, Issue 4, July-August 2026
Published On	2026-08-04
DOI	https://doi.org/10.36948/ijfmr.2026.v08i04.85021

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Reconstructing the Prime Zeta Function from Semiprimes: A Stable Nested Radical and Its Condition Number.

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