(ZPP) Analytic Framework for Zeta Functions of Polynomial-Generated Prime Sequences: Proposal and Initial Results

This deposit bundles a comprehensive analytic framework for studying zeta functions associated with polynomial-generated prime sequences. All files are provided as both editable .tex sources and compiled .pdf outputs. Contents Proposal Title: Analytic Framework for Zeta Functions of Polynomial-Generated Prime Sequences Summary: Defines a new Dirichlet series Z_f(s) = sum_{n>=1} [f(n) prime] / |f(n)|^s, formalizes “prime order” (the maximal run of consecutive prime values), and outlines background, objectives, methodology, and expected applications in number theory and cryptography. Research Result 1 Title: Absolute Convergence of Z_f(s) Main Theorem: Z_f(s) converges absolutely whenever Re(s) > 1/deg(f). Proof Strategy: Comparison test against a p-series and Landau’s indicator formula. Research Result 2 Title: Singularity Analysis of Z_f(s) at s = 1/deg(f) Conjecture: Under Bateman–Horn, Z_f(s) admits a simple pole at s = 1/deg(f) with residue C_f/deg(f), where C_f is the Bateman–Horn constant. Derivation: Heuristic via partial summation and Mellin arguments. Research Result 3 Title: Rigorous Justification of Mellin Inversion and Tauberian Framework Outline: A contour-deformation argument justifying the inversion of the Mellin transform for Z_f(s). A weighted Tauberian theorem converting the simple pole at s = 1/deg(f) into the precise prime-count asymptoticπ_f(X) ~ (C_f/deg(f))·X / log X. Model Polynomial Selection Title: Recommended Model Polynomials and Selection Rationale Benchmark List: f(x)=x (deg = 1; C_f=1; residue at s=1 equals 1) f(x)=x²+x+41 (deg = 2; C_f≈1; residue at s=1/2≈0.5) f(x)=x²+1 (deg = 2; C_f≈0.6864; residue≈0.3432) f(x)=2x+1 (deg = 1; C_f=2; residue at s=1 equals 2) f(x)=x³+2 (deg = 3; C_f≈0.7788; residue at s=1/3≈0.2596) Purpose: These cover linear, quadratic, and cubic cases with both trivial and nontrivial C_f, providing ideal testbeds for numerical residue estimation.