Upper bound for the integral over minor arcs of the exponential sum with prime omega function coefficients

Question

Define $\mathfrak{m}$ as the union of the minor arcs of the form $|\alpha-\frac{a}{q}|\leq 1/qQ$, with $(a,q)=1$ and $Q_0<q\leq Q$, with $Q_0\geq N/Q$, for a certain $N\geq Q$ large.

Is it possible to find a sharp upper bound for
$$\int_{\mathfrak{m}}|\sum_{n\leq N}\omega(n)e(n\beta)|^2d\beta$$

for the additive function $\omega(n)$, which counts the number of distinct prime factors of an integer $n$?

For sharp upper bound I mean nothing that uses Parseval's identity. The idea should be to insert the definition of $\omega(n)$ in the exponential sum and then use the structure of the minor arcs in question to obtain a saving compared to the straightforward application of Parseval's identity.

Thanks in advance!

Answer 1

Not an answer just a small comment: if the singular series coming from the problem of evaluating $$\sum _{n-n=0}h(n)h(n')$$ with the circle method is equal to $$\int _0^1\left |\sum_{n\leq N}h(n)e(n\beta)\right |^2d\beta$$ then you can write your minor arc contribution as $$\left (\int _0^1-\int _\mathfrak M\right )\left |\sum_{n\leq N}h(n)e(n\beta)\right |^2d\beta ,$$ so you just need to do the major arcs. (Maybe you know this already, but just in case.)