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arXiv · 2609.25341

Distinct exponents in the prime factorization

Mikhail R. Gabdullin·Vitalii V. Iudelevich
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Abstract

Following Erdős (1982) and Sanna (2019), we study the arithmetic function $h(n)$, which is defined to be the number of distinct exponents in the prime factorization of a positive integer $n$. Among other things, we show that $$ \sum_{n\leq x}h(ϕ(n)) \asymp x\left(\frac{\log\log x}{\log\log\log x}\right)^{1/2}, $$ where $ϕ$ is the Euler totient function. The key ingredient is the Poisson random model for $ω(n,T)$, the number of the prime divisors of $n$ in a given subset of primes $T$, which was introduced in a recent work of Ford.

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Mikhail R. Gabdullin, Vitalii V. Iudelevich. 2026-09-21. Distinct exponents in the prime factorization. https://arxiv.org/abs/2609.25341

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