[จูกัดเหลียง]

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Prove that \(\displaystyle \sum_{n\le x}\varphi(n)\left\{\,\dfrac{x}{n}\right\}=\dfrac{x^2}{\zeta(2)}-\dfrac{\left[\,x\right](\left[\,x\right] +1) }{2} +O(\log x)\)
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Prove that \(\displaystyle \sum_{n\le x}\left\{\,\dfrac{x}{n}\right\} =(1-\gamma)x+O(\sqrt{x})\)
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Prove that \(\displaystyle \sum_{\substack{1\le k\le n\\\gcd(k,n)=1}} k=\dfrac{n}{2}\varphi(n)\)
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Prove that \(\displaystyle\prod_{d|n}d=n^{\tau(n)/2}\)
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Prove that \(\displaystyle\sum_{n\le x}\dfrac{\tau(n)}{n}=\dfrac{1}{2}\log^2 x+2\gamma\log x+(\gamma^2-\gamma_1)+O\left(\dfrac{\log x}{x}\right)\)
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Prove that \(\displaystyle\sum_{n\le x} \dfrac{\mu(n)}{n^2}=\dfrac{1}{\zeta(2)}+O\left(\dfrac{\log x}{x}\right)\Longrightarrow \sum_{n\ge 1} \dfrac{\mu(n)}{n^2}=\dfrac{1}{\zeta(2)}\)
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