On the Hardy-Ramanujan Theorem

Benjamin Durkan

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arXiv · 2310.14760

On the Hardy-Ramanujan Theorem

Benjamin Durkan

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Abstract

In this note we prove an effective version of the Hardy--Ramanujan Theorem. For every $x\ge 2$ and every non-negative function $F$ on the non-negative integers, we show $$\frac{1}{x}\sum_{2\le n\le x}F(\omega(n)-1)\le 118\,\mathbb{E}F(Z_{\log\log x+4.096}),$$ where $Z_{\lambda}$ is Poisson with parameter $\lambda$. Thus the shifted empirical distribution of $\omega(n)$ is pointwise dominated by a fixed multiple of a Poisson law. We also obtain the sharper squarefree analogue, derive explicit Chernoff and Gaussian-window estimates, obtain moderate-deviation upper bounds and uniform moment estimates, and transfer these consequences to $\Omega(n)$ and to the number of prime divisors occurring exactly once.

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Benjamin Durkan. 2023-10-23. On the Hardy-Ramanujan Theorem. https://arxiv.org/abs/2310.14760

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