I was playing with Wolfram Alpha online calculator when I wondered what about the asymptotic behaviour as $x\to\infty $ of $$\int_1^x\sin^2(\pi(t))dt,\tag{1}$$ where $\pi(x)$ denotes the prime-counting function, and the integrand is the square of such sine function, that is $\sin^2(\pi(u))$ means $(\sin(\pi(u)))^2$. Here you've these codes
plot (sin(primepi(x)))^2, for 1<x<100
plot (sin(primepi(x)))^2, for 1<x<10000
versus
plot (sin(x/log(x)))^2, for 2<x<10000
I think that study the asymptotic behaviour of $(1)$ is very difficult, thus I ask this question about if you can provide me calculations for an example.
Question. What work can be done to study the area under the graph of $f(y)=\sin^2(\pi(y))$ over a segment $[N,M]$, where $N<M$ are large positive integers, that is the behaviour of an example* $$\int_N^M\sin^2(\pi(t))dt?\tag{2}$$ *To me only is required the analysis of an example for some large $N<M$. Many thanks.
That I am asking is if you can do some work about it.