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arXiv · 2304.04805

Karatsuba's divisor problem and related questions

Mikhail R. Gabdullin·Vitalii V. Iudelevich·Sergei V. Konyagin
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Abstract

We prove that $$ \sum_{p \leq x} \frac{1}{\tau(p-1)} \asymp \frac{x}{(\log x)^{3/2}}, \quad \quad \sum_{n \leq x} \frac{1}{\tau(n^2+1)} \asymp \frac{x}{(\log x)^{1/2}}, $$ where $\tau(n)=\sum_{d|n}1$ is the number of divisors of $n$, and the summation in the first sum is over primes.

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Mikhail R. Gabdullin, Vitalii V. Iudelevich, Sergei V. Konyagin. 2023-04-10. Karatsuba's divisor problem and related questions. https://arxiv.org/abs/2304.04805

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