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

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 $$Q_0, with $$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.

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.)