Chinmay has had the idea of a function for some time, today we were able to get some stuff off of it.

we define (Chinmay's definition): \[Ch(n)=\begin{cases} 1&, n=\text{ prime} \\ (-1)^{\tau(n)}&,\text{ otherwise}\end{cases}\] Later, I made it to \[Ch(n)=\begin{cases} -1&, n=\text{perfect square}\\ 1&,\text{otherwise}\end{cases}\] this is trivial, (tau is odd iff \(n\) is a perfect square, tau of all primes are 2). let's find its Dirichlet series \[D(Ch;s)=-2\zeta(2s)+\zeta(s)\] This is an interesting result, we will try to apply it in future notes.

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