In the case of a monotonic function \(f\), for \(\sum_{p \le n} f(n)\), we estimate as follows: \[\sum_{p \le n} f(n) = \int_{1^-}^n f(t) \, \mathrm{d}\pi(t) = \pi(n) f(n) - \int_1^n \pi(t) f'(t) \, \mathrm{d}t,\] then we can apply the PNT.
As for multiplicative functions \(f\), namely those of the form \(f(n) = \chi(n) n^{-s}\), we can estimate \(\sum_{p \le n} f(n)\) by using the zero-free region of \(L(s, \chi)\).
In another case where \(f\) is oscillatory but not multiplicative, for instance \(f(n) = \mathrm{e(n\alpha)}\), we apply the philosophy of estimation over the bilinear form.
In 1977, Vaughan found a way of dealing with \[\sum_{n \le x} \Lambda(n) f(n),\] which gave a bilinear form of its identity. Let \(u > 0\), \(n > 0\), \(y \ge z\), and define \[F(s) = \sum_{n \le u} n^{-s}, \qquad G(s) = \sum_{m \le v} \mu(m) m^{-s}.\] Then \[\sum_{n} \Lambda(n) f(n) = S_1 - S_2 - S_3 + S_4,\] where \[\begin{aligned} &S_1 = \sum_{n \le u} \mu(n) \sum_{n \le m \le x/n} (\log n) f(mn),\\ &S_2 = \sum_{m \le u} c_m \sum_{n \le x/m} f(mn), \quad c_m = \sum_{\substack{k \le u \\ \ell \le v}} \Lambda(k) \mu(\ell),\\ &S_3 = \sum_{m \le u} \sum_{\substack{n \le y \\ mn \le x}} \Lambda(m) \left( \sum_{\ell \le u} \mu(\ell) \right) f(\ell mn),\\ &S_4 = \sum_{n \le u} \Lambda(n) f(n).\end{aligned}\]
This is a direct result followed from the coefficient of \(n^{-s}\) of the following identity: \[-\frac{\zeta'(s)}{\zeta(s)} = G(s) \left( -\frac{\zeta'(s)}{\zeta(s)} \right) - F(s) G(s) \frac{\zeta'(s)}{\zeta(s)} - \left( -\frac{\zeta'(s)}{\zeta(s)} - F(s) \right) \left( G(s) \frac{\zeta'(s)}{\zeta(s)} - 1 \right) + F(s).\]
In applications, we shall bound \(S_4\) trivially.
For \(S_2\), note that \[|c_m| = \left| \sum_{\substack{k \le u, \ \ell \le v \\ k\ell = m}} \Lambda(k) \mu(\ell) \right| \ \le\ \sum_{\ell \mid m} \Lambda(\ell) = \log m,\] then it follows that \[S_2 \ \ll\ (\log uv) \sum_{m \le u} \left| \sum_{\substack{n \le x/m}} f(mn) \right|.\] For \(S_1\), we write \[S_1 = \sum_{m \le u} \mu(m) \sum_{\substack{n \le x/m}} f(mn) \int_1^n \frac{\mathrm{d}t}{t} = \int_1^x \sum_{m \le u} \mu(m) \sum_{\substack{t \le n \le x/m}} f(mn) \frac{\mathrm{d}t}{t}\] \[\ll\ C (\log qy) \sum_{m \le u} \max_{t} \sum_{\substack{t \le n \le x/m}} f(mn).\]
If we have an inequality of the sort \[\left| \sum_{m,n} a_m b_n f(mn) \right| \ \le\ \Delta \left( \sum_{m} |a_m|^2 \right)^{1/2} \left( \sum_{n} |b_n|^2 \right)^{1/2},\] where \(\Delta\) depends on the coefficient matrix \(F = [f(mn)]\), independent of the vectors \(a, b\).
Then, for \(S_3\), \[S_3 \ \ll\ (\log qy) \left( \max_{u < m \le 2u} \Delta \left( \sum_{m \le 2u} \Lambda(m)^2 \right)^{1/2} \left( \sum_{n \le qy/m} d(n)^2 \right)^{1/2} \right)\] \[\ \ll\ y^{1/2} (\log qy)^3 \max_{u < m \le 2u} \Delta.\]