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: Sheng Li (gmachine1729) wrote,
2017-10-10 04:33:00

Sheng Li
gmachine1729
2017-10-10 04:33:00

Asymptotic formula for square free integers

Originally published at 狗和留美者不得入内. You can comment here or there.

$\begin{aligned} \displaystyle\sum_{n \leq x} \displaystyle\sum_{d^2 | n} \mu(d) & = \displaystyle\sum_{d \leq \sqrt{x}} \mu(d)\left\lfloor \frac{x}{d^2} \right\rfloor \\ & = x\displaystyle\sum_{d \leq \sqrt{x}} \frac{\mu(d)}{d^2} + O(\sqrt{x}) \\ & = x \frac{6}{\pi^2} + O(x\displaystyle\sum_{d > \sqrt{x}} \frac{1}{d^2} + \sqrt{x}) \\ & = x \frac{6}{\pi^2} + O((1 + \sqrt{x}) + \sqrt{x}) \\ & = x \frac{6}{\pi^2} + O(\sqrt{x}). \end{aligned}$

Tags: analytic number theory, number theory

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Asymptotic formula for square free integers

中国姓氏的南北分布

Why I think Stanford professor James Landay's piece on computing in China in 2011 exposed his idiocy

China Will Overtake the US in Computing…Maybe, Someday… by Stanford professor James Landay in 2011

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