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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2509.04102 |
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| _version_ | 1866918135755440128 |
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| author | Gopal, Mantha Sai |
| author_facet | Gopal, Mantha Sai |
| contents | The Erdos-Kac theorem, a foundational result in probabilistic number theory, states that the number of prime factors of an integer follows a Gaussian distribution. In this paper we develop and analyze probabilistic models for "random integers" to study the mechanisms underlying this theorem. We begin with a simple model, where each prime p is chosen as a divisor with probability 1/p in a sequence of independent trials. A preliminary analysis shows that this construction almost surely yields an integer with infinitely many prime factors. To create a tractable framework, we study a truncated version Nx = product of p<=x of p^Xp, where Xp are independent Bernoulli(1/p) random variables. We prove an analogue of the Erdos-Kac theorem within this framework, showing that the number of prime factors omega(Nx) satisfies a central limit theorem with mean and variance asymptotic to log log x. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_04102 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Probabilistic Framework for the Erdos-Kac Theorem Gopal, Mantha Sai General Mathematics The Erdos-Kac theorem, a foundational result in probabilistic number theory, states that the number of prime factors of an integer follows a Gaussian distribution. In this paper we develop and analyze probabilistic models for "random integers" to study the mechanisms underlying this theorem. We begin with a simple model, where each prime p is chosen as a divisor with probability 1/p in a sequence of independent trials. A preliminary analysis shows that this construction almost surely yields an integer with infinitely many prime factors. To create a tractable framework, we study a truncated version Nx = product of p<=x of p^Xp, where Xp are independent Bernoulli(1/p) random variables. We prove an analogue of the Erdos-Kac theorem within this framework, showing that the number of prime factors omega(Nx) satisfies a central limit theorem with mean and variance asymptotic to log log x. |
| title | A Probabilistic Framework for the Erdos-Kac Theorem |
| topic | General Mathematics |
| url | https://arxiv.org/abs/2509.04102 |