I find "closed formula" for divisors of an positive integer, it is useful or not?

Question

I find this formula, but i don't know this is worth or not. $$d(n)=\sum_{i=1}^{n} \lim_{j\to\infty} (cos\left(\frac{\pi n}{i} \right))^{2j}$$
It is possible to improve it to deduce formula for $P_n$ or $\pi(n)$?

Answer 1

No, that kind thing is not valuable -- at least it's not "valuable" in the sense that anyone is going to reward you for thinking it up.

First, it is at best debatable whether what you have there is a "closed formula". A finite summation may arguably count as "closed" in some contexts, but the limit is certainly not what people think of when they speak of "closed formulas".

(It is "closed" in the logical sense that it doesn't contain free variables other than $n$, but that is not the meaning of "closed" that is relevant here).

More to the point, the interest in having "closed formulas" for this and that is that the closed formulas are supposed to be more efficient to calculate and reason about than other more direct ways of defining the thing you're interested in. But while your formula looks like it has the right value, it is certainly much more cumbersome to apply than the trivial description "try all the integers from 1 up to $n$, divide $n$ by each of them, and count how many times the division comes out even". On the contrary, even evaluating your formula in the most straightforward way requires a good amount of theory for figuring out error bounds for the limit such that one knows how far to take $j$ to get an acceptable evaluation.

In principle, it there were some kind of useful symbolic manipulation one could do with your formula that can't be done just as straightforwardly with the trivial description (or, say, one that uses an Iverson bracket or the indicator function of $\mathbb Z$ instead of that funky trigonometric limit), then you might be on the trace of something. But the onus would be on you to show what that useful manipulation would be.

Answer 2

It's Beautiful, but it seem's not useful. because we have no appropriate finite approximate for $\lim_{j\to\infty} (cos\left(\frac{\pi n}{i} \right))^{2j}$ in your formula.