Let $\omega(n)$ count the number of distinct prime factors of the integer $ n > 2$. This $\omega(n)$ is called the prime omega function.
Consider
$$ \beta_a^b(n) = \sum_{i=2}^n (-a)^{\omega(i)} \space \omega(i)^b $$
In particular
$$ \beta_a^0(n) $$
And
$$ \beta_1^b(n) $$
For $a,b > 0 $
How do these behave ?
How are the asymptotics ? What is known and what is proven ?
I assume $ \beta_a^0(n) $ behaves like
$$ O( (1 - \ln^{\alpha a + \gamma }(n) ) \space n ) $$
For Some constants $\alpha , \gamma $ independant of $n,a$.
And $ \beta_1^b(n) $ behaves like
$$ n^b \space rw(n) $$
Where $$rw(n) $$ means a random walk : adding $n $ elements randomly chosen from ${-1,1}$.
Many will note the similarities with more traditional functions like e.g. liouville or Mertens’ function.
I therefore considered connections with dirichlet series and extending to the complex plane but i think that is not a fruitful approach in this case , right ?
I considered using ramanujan sums and related sums.
Terry Tao ( to call a big name ;) ) talked about the parity problem. And i guess this relates when i think of it in terms of sieves.
Papers considering this - or not too different - would be Nice too.
I know the Mertens’ conjecture ( about Mertens’ function ) failed and Many related questions are still open and therefore assume this question is nontrivial.
——
Finally i assume non-universal 1D cellular automatons are not related to this but feel free to show me wrong. This is not the main question ofcourse.
——