A Smooth Analytical Approximation of the Prime Characteristic Function
Stanislav Semenov
TL;DR
The paper introduces a smooth, differentiable surrogate $P(n)\in[0,1]$ for the prime indicator $\chi_{\mathrm{prime}}(n)$, defined via a triple integral with a periodic kernel that detects approximate integer ratios $x(t)/y(u,v)$. It proves both a constructive finite-interval version and an asymptotic limit version, showing that $P(n)\to1$ for primes and $P(n)<1$ for composites under appropriate tuning of $(\delta,\varepsilon,p)$, with $P(n)$ being $C^\infty$. The work develops a family of variants (summed integral, one-dimensional reductions, localized bell localization) to enhance divisibility signaling while maintaining smoothness, and it analyzes convergence, error bounds, and scaling with $n$. It further discusses computational aspects, including numerical integration strategies, complexity, and potential data-driven approaches, and outlines diverse applications in spectral analysis, optimization, ML, and cryptography. By providing both finite-constructive and asymptotic perspectives, the paper offers a new analytic framework that blends continuous methods with discrete primality structure, enabling gradient-based, probabilistic, and visualization-based explorations of primes.
Abstract
We construct a smooth real-valued function P(n) in [0,1], defined via a triple integral with a periodic kernel, that approximates the characteristic function of prime numbers. The function is built to suppress when n is divisible by some m < n, and to remain close to 1 otherwise. We prove that P(n) approaches 1 for prime n and P(n) is less than 1 for composite n, under appropriate limits of the smoothing parameters. The construction is fully differentiable and admits both asymptotic and finite approximations, offering a continuous surrogate for primality that is compatible with analytical, numerical, and optimization methods. We compare our approach with classical number-theoretic techniques, explore its computational aspects, and suggest potential applications in spectral analysis, machine learning, and probabilistic models of primes.