An excerpt from Introduction to Analytic Number Theory by Tom M. Apostol. I have three main concerns regarding this proof: What is the abscissa of convergence for $\frac{1}{F(s)}$ – why can we take the product of the two series. I only know this is true if the two series are absolutely convergent. Why can we […]

I’m trying to work through Ireland and Rosen’s A Classical Introduction to Modern Number Theory as I’ve heard good things about it. This is Exercise 12 from Chapter 2. Here $\mu$ is the Moebius function, and $\phi$ the totient function. Find formulas for $\sum_{d|n}\mu(d)\phi(d)$, $\sum_{d|n}\mu(d)^2\phi(d)^2$, and $\sum_{d|n}\mu(d)/\phi(d)$. Playing around with the first sum, I know […]

Is there any closed form expression for $\displaystyle\sum_{d|n} d\phi(d)$? I have tried a lot but can only reduce it to $\displaystyle\sum_{k=1}^{n}\frac{n}{(k,n)}$ where $(k,n)$ is the greatest common divisor. But it cannot be simplified.

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