TITLE: Arithmetical Functions Sum, $\sum\limits_{d|n}\sigma(d)\phi(\frac{n}{d})$ and $\sum\limits_{d|n}\tau(d)\phi(\frac{n}{d})$
QUESTION [8 upvotes]: $$\sum_{d|n}\sigma(d)\phi\left(\frac{n}{d}\right)=n\tau(n)
,\\ \sum_{d|n}\tau(d)\phi\left(\frac{n}{d}\right)=\sigma(n)$$
The problem (7.4.15) of Burton's Elementary Number Theory has been request to prove the above equalities. In this book Dirichlet multiplication or Riemann's zeta function isn't expressed before this problem. In addition, I know that the first time, Pillai has been proved this equalities but I couldn't find the Pillai's paper on the web. Can you please refer me to a link of Pillai's paper? or give me a proof without the use of Dirichlet multiplication or Riemann's zeta function?

REPLY [4 votes]: There is a request for Pillai's paper or proof. The paper is a bit hard to find and quite old, so I scanned it. It's called "On An Arithmetic Function" and was published at Annamalai University.
I've made the scan available at my website. [Presumably I'll host it somewhere else sometime - I do not know of another online copy]. The first desired equation is the combination of his Theorem II and Theorem III, for instance. The second desired equation isn't an explicit result, but instead is a byproduct of his other proofs and results.