Anyway, by quoting Lubos Motl's blog post of 2018/19, i'm lead to guess you are talking about the integrals on the extreme right-hand side of (13) and (14). Indeed, the author clearly and rigorously showed that those integrals are absolutely convergent and real-analytic for sigma > Theta, such that f(sigma) and g(sigma) both have real- analytic continuations there.
In the 2018 version that was reviewed by Lubos, the author had an equation of the form
A(s) = B(s)
for Re(s)>1, where A(s) is some improper integral and B(s) is some function that is analytic in some (larger) half-plane. From this, one can only deduce that A(s) has an analytic continuation to the larger plane, but one cannot deduce (as the author did in that 2018 version) that A(s) converges there.
For example, let A(s) be the integral of x^{-s} w.r.t. x on [1, infty). Then B(s) = (s-1)^{-1} for Re(s) > 1. Notice that B(s) hence A(s) has a meromorphic continuation to the entire complex plane with a (simple) pole only at s=1. However, one cannot deduce from this that A(s) converges for some s with Re(s) < 1. The mistake in the author's previous version is analogous to saying that A(s) converges for Re(s) < 1. However, to his credit, he seems to have been working hard to eliminate these kinds of mistakes.
I guess that's what he means on his arXiv page when he said, "though all of my previous proposed (dis)proofs were flawed, i think the flaws pointed me towards the right direction".
Credit where it's due.