  In this paper, we prove a conjecture by Gukov-Pei-Putrov-Vafa for a wide
class of plumbed 3-manifolds. Their conjecture states that
Witten-Reshetikhin-Turaev (WRT) invariants are radial limits of homological
blocks, which are $ q $-series introduced by them for plumbed 3-manifolds with
negative definite linking matrices. The most difficult point in our proof is to
prove the vanishing of weighted Gauss sums that appear in coefficients of
negative degree in asymptotic expansions of homological blocks. To deal with
it, we develop a new technique for asymptotic expansions, which enables us to
compare asymptotic expansions of rational functions and false theta functions
related to WRT invariants and homological blocks, respectively. In our
technique, our vanishing results follow from holomorphy of such rational
functions.
