Let $\omega(n)$ count the number of distinct prime factors of the integer $ n \geq 2$. This $\omega(n)$ is called the prime omega function.
Inspired by these ideas :
Improved sieve for primes and prime twins?
and the idea that the prime twin counting function is close to
$$ \pi_2(x) \sim x \prod_{n=2}^{\sqrt x+2} (1 - 2/p_n) $$
$$\pi_2(x)\sim 2\Pi_2 \frac{x}{\log^2(x)},$$ where $\pi_2$ denotes the number of twin primes smaller than $x\in[0,\infty)$
the following question naturally occurs :
$\pi_2(t^2) =^? \sum_{2<j<t^2} (-2)^{\omega(j)} (1/2)(\lfloor{\frac{t^2}{j}}\rfloor +\lfloor{\frac{t^2-2}{j}}\rfloor)$ where $j $ are squarefree odd integers.
Is this formula correct or correct upto some fixed constant addition $C$ ?
OR maybe this one :
$\pi_2(t^2) - \pi_2(t) =^? \sum_{2<j<t^2} (-2)^{\omega(j)} (1/2)(\lfloor{\frac{t^2}{j}}\rfloor +\lfloor{\frac{t^2-2}{j}}\rfloor)$ where $j $ are squarefree odd integers.