Soft Arithmetic: A Parameterized Framework for Generalized Divisibility and the Riemann Hypothesis

We introduce Soft Arithmetic, a novel parameterized arithmetic system based on a continuous measure of divisibility δ#(d, n; γ). This framework generalizes classical integer arithmetic by allowing smooth transitions between divisibility and non-divisibility, governed by a tunable stiffness parameter γ. We construct the corresponding soft zeta function ζ#(s; γ), establish its meromorphic properties, and propose a Soft Riemann Hypothesis asserting that its quasi-zeros lie along a parameterized critical line ℜ(s) = ρ(γ). Additionally, we formulate soft versions of Goldbach’s, twin primes, and Legendre’s conjectures using regularized prime indicator functions. This approach offers a new lens for number theory, resilient to noise and applicable in both theoretical and data-driven contexts.