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\varphi summation

Let \varphi denote the Euler’s function. Prove that

    \[\sum_{k=1}^{\infty} \varphi(k) \left \lfloor \frac{n}{k} \right \rfloor = \frac{n \left ( n+1 \right )}{2}\]

Solution

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Convergence of series

Let \mathrm{gpf} denote the greatest prime factor of n. For example \mathrm{gpf}(17) = 17 , \mathrm{gpf} (18) =3. Define \mathrm{gpf}(1)=1. Examine if the sum

    \[\mathcal{S} = \sum_{n=1}^{\infty} \frac{1}{n \; \mathrm{gpf}(n)}\]

converges.

Solution

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Sigma divisor sum

Let \sigma denote the divisor function. Prove that

    \[\lim_{n \rightarrow +\infty} \frac{1}{n^2} \sum_{k=1}^{n} \sigma(k) = \frac{\pi^2}{12}\]

Solution

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On multiplicative functions

Let \mu denote the Möbius function and \varphi the Euler’s totient function. Prove that

    \[\frac{n}{\varphi(n)}=\sum_{d \mid n} \frac{\mu^2(d)}{\varphi(d)}\]

Solution

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On the factorial

Let \mu denote the Möbius function and \left \lfloor \cdot \right \rfloor denote the floor function. Prove that:

    \[n! = \prod_{j=1}^{\infty} \prod_{i=1}^{\infty} \left ( \left \lfloor \frac{n}{ij} \right \rfloor! \right )^{\mu(i)}\]

Solution

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