In a paper by C. P. Willans (Here) ; He Introduced a following elementary formula for prime counting function .
$$\pi (n)=\sum_{j=2}^{n}\frac{\sin^{2}\left(\pi \frac{(j-1)!^{2}}{j}\right)}{\sin^{2}(\frac{\pi }{j})}$$
I also read in some paper that these type of formulas don't have a practical use (?) . So I tried to justify it by following observation and/or difficulties:
As we can see the $!$ can be replaced by $\Gamma$ .
(1)So, the $\frac{\sin^{2}\left(\pi \frac{(j-1)!^{2}}{j}\right)}{\sin^{2}(\frac{\pi }{j})}$ (summand ) is analytic for $j>0$ ( we can include zero in the domain by shifting the $j$ in summand by 1).
(2)The summand is in purely distributional sense
(3) Also , It oscillate violently in Right half plane .( So criteria for most of the analytic summation formulas ruled out ).
(More possible difficulties?)
Question :
Although with the given difficulties , summand is analytic in non negative domain ; So , is there any way to get an asymptotic estimate for prime counting function using this formula ?
Are there any other (stronger) difficulties than mentioned above which are obstructing the formula to be analysed asymptotically ?
At least can we expect to prove infinitude of primes using this ? (If not what's the reason for it's impenetrable nature )?