MathOverflow

While trying to bound the number of permutations avoiding a certain family of consecutive patterns, I obtained a continued fraction that converges in $\mathbb C[[z]]$ to a related OGF $\omega(z)$. The EGF for permutations avoiding the patterns is $\frac...

I read the Hatcher's book (https://pi.math.cornell.edu/~hatcher/AT/AT.pdf).
Let $M$ be an orientable genus 2 surface.
In Example 3.31 (page 241), there are two statements I have been stacked.

I would like to apologize in advance for a too technical question. Let us consider the following fourth order polynomial equation in $x$:\begin{eqnarray} F(x) \equiv (2 a p +2)x^4+ (6a(1-a)p^2+(6-12a)p-6)x^3 \nonumber \\ + p(2(a-2)(a-1)a p^2 + 3(5a^...

Let $0 < \delta < \Delta < 1/2$. Is it possible to prove something like the statement below?It does not seem to follow directly from the statement of Heath-Brown's identity.Any comments or reference would be appreciated! Thank you!
Statement: There exists an integer $k = k(\delta, \Delta)$ such that for any $X \ge 2$ and any arithmetic function $f$ supported on $(X/2, X]$, one can write$$\sum_{ n \in \mathbb{Z}} \Lambda(n)\,f(n)$$as a sum of $O((\log X)^C)$ sums of the form$$\sum...