2025/06/22 by Fuchs, Sebastian
Agricultural and Biological Sciences · Economics, Econometrics and Finance · #Agriculture and Rural Development Research #FOS: Mathematics #Italy: Economic History and Contemporary Issues #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2506.18933
openalex publication_date 2025/06/22 · openalex created_date 2025/10/14 · openalex updated_date 2026/08/01
A C1 prime indicator P\colonℝ→ℝ is constructed by applying the Fejér identity to the sine-quotient encoder of trial division. For integers n≥ 2, \mathcal P(n)=0 holds exactly for odd primes; \mathcal P(2)>0. For all non-integers x>1 one has \mathcal P(x)>0. The function is piecewise C^∞ and its second derivative has jumps precisely at the squares m2, with explicit sizes. Replacing the sharp cut-off by a smooth transition yields C^∞ analogues Pτ and Pσ with integer limits Pτ(n;κ)→ τ(n)-2 and Pσ(n;κ)→ σ(n)-n-1 as κ→∞, obtained from locally uniform convergence of derivative series. For large κ, numerical evidence indicates companion zeros near odd primes for Pτ and an asymmetric pair for Pσ. No assertion is made beyond integer input, and no statements are claimed about the prime number theorem or zero distributions of L-functions. The appendix includes two illustrative prime-counting sums.