2025/06/22 by Fuchs, Sebastian · 1 citation
Mathematics · #FOS: Mathematics #Functional Equations Stability Results #General Mathematics (math.GM) #Mathematical Dynamics and Fractals #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2509.12297
openalex publication_date 2025/06/22 · openalex created_date 2025/10/18 · openalex updated_date 2026/07/28
A Fejér-Dirichlet lift is developed that turns divisor information at the integers into entire interpolants with explicit Dirichlet-series factorizations. For absolutely summable weights the lift interpolates (a*1)(n) at each integer n and has Dirichlet series ζ(s)A(s) on \Re s>1. Two applications are emphasized. First, for q>1 an entire function \mathfrak F(⋅,q) is constructed that vanishes at primes and is positive at composite integers; a tangent-matched variant \mathfrak F\sharp is shown to admit an explicit, effective threshold P0(q) such that for every odd prime p≥ P0(q) the interval (p-1,p) is free of real zeros and x=p is a boundary zero of multiplicity two. Second, a renormalized lift for a=μ*Λ produces an entire interpolant of Λ(n) and provides a constructive viewpoint on the appearance of ζ'(s)/ζ(s) through the FD-lift spectrum. A Polylog-Zeta factorization for the geometric-weight case links ζ(s) with Lis(1/q). All prime/composite statements concern integer arguments. Scripts reproducing figures and numerical checks are provided in a public repository with an archival snapshot.