I found the answer from Zeb Brady. The relevant method is the Selberg-Delange method, see for example this paper. In particular Selberg proved that the average order of $z^{\Omega(n)}$, where $|z| < 2$ is not $-1$ (when it would be a pole of $\Gamma(z)$) is estimated by
$$ \sum_{n\le x} z^{\Omega(n)} = (\frac{f(1,z)}{\Gamma(z)}+o(1)) x (\log x)^{z-1} $$
where $f$ is some convergent Euler product.
So for $z$ any root of unity other than $-1$ we get a comparatively huge main term (as Terry's comments predict). Maybe other people can comment on where the $1/\Gamma(z)$ comes from and how it allows Mobius to cancel unlike everything else.
As suggested in the comment by Stanley Yao Xiao, Selberg's choice of $\lambda_d$ in his clever upper bound sieve is an application of a "discrete" variational principle. This answer is going to present another concept arising from sieve theory that may possibly be connected with physics.
In physics, objects are studied via differential equations, and in analytic number theory theory, sieves can be studied via differential-difference equations.
In 1964, Ankeny and Onishi proved that if $\sigma(u)$ is the solution to the IVP that
$$
\sigma(u)={e^{-\gamma\kappa}\over\Gamma(\kappa+1)}\left(\frac u2\right)^\kappa\quad(0<u\le2),
$$
$$
[u^{-\kappa}\sigma(u)]'=-\kappa u^{-\kappa-1}\sigma(u-2)\quad(u>2),
$$
then we have
Theorem (Ankeny & Onishi, 1964; Halberstam & Richert, 1974): For $u>0$, if $\nu(d)$ denote the number of distinct prime divisors of $d$, $\omega(d)$ is a multiplicative function and $X,R_d$ are numbers satisfying
$$
|\mathcal A_d|={\omega(d)\over d}X+R_d,
$$
$$
-L\le\sum_{w\le p<z}{\omega(p)\log p\over p}-\kappa\log\frac zw\le O(1),
$$
then
$$
S(\mathcal A,\mathcal P,z)<X\prod_{p<z}\left(1-{\omega(p)\over p}\right)\left\{{1\over\sigma(u)}+O\left(L\cdot{u^{2\kappa}+u^{-\kappa-1}\over\log z}\right)\right\}+\sum_{\substack{p|d\Rightarrow p<z\\d<z^u}}3^{\nu(d)}|R_d|
$$
Best Answer
Sieve theory is not saturated. It is alive and thriving. The ICM just awarded the Fields medal to James Maynard in no small part due to his work in sieve theory (see here and here). Because of the natural role of multiplicative structure in sieve theory, there are lots of fascinating and important results in the theory of $L$-functions and modular forms that rely decisively on sieve theory. In these instances, it is usually the flexibility of sieve methods that ends up being the key to success. On the other hand, there are lots of things that one can study about $L$-functions and modular forms that have nothing to do with sieve theory.
For a first look at sieve theory, I think that "An Introduction to Sieve Methods and Their Applications" by Cojocaru and Murty is very nice. As Stanley Xiao mentioned above, a more advanced book (but still nice to read) is "Opera de Cribro" by Friedlander and Iwaniec.
Here are some examples where sieve theory, $L$-functions, modular forms, and other ideas combine, resulting in truly splendid results. This list is far from exhaustive.