A Smooth Analytical Approximation of the Prime Characteristic Function
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866915307025596416 |
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| author | Semenov, Stanislav |
| author_facet | Semenov, Stanislav |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_14414 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Smooth Analytical Approximation of the Prime Characteristic Function Semenov, Stanislav General Mathematics 11A41, 11Y16, 26E40, 03F60 F.4.1 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. |
| title | A Smooth Analytical Approximation of the Prime Characteristic Function |
| topic | General Mathematics 11A41, 11Y16, 26E40, 03F60 F.4.1 |
| url | https://arxiv.org/abs/2504.14414 |