I can understand how the verification of hypotheseses based on inductive reasoning can be problematic, and I understand that a lot of prominent figures in philosophy are in opposition to this (Popper, for example).
But it is not entirely clear to me whether the problem is induction or inductivism? By 'induction', I mean, for example, that we observe that the non-trivial zeros of the Riemann Zeta function all lie in a certain interval, and thus we propose the hypothesis that all non-trivial zeros of this function lie in that interval... and then we can attempt to verify or falsify that hypothesis based on other scientific methods (that is, not by simply finding more zeros in the same interval which would be inductivism). That is, induction is only part of the chain, the part that leads to the formulation of a hypotheses (which can then be studied in more appropiate manners).
Is this latter kind of thinking also grouped along with the inductivism that is opposed by many? If yes, how would they then come up with their hypotheses to begin with (there are other ways, yes, but induction seems to be the most obvious one)?