In the MO-question on a lacunary series I wrote an answer (really: a comment) having the hypothese about an interesting fact (the function $d(x)$ as difference between the OP's function $f(x)$ and some modification $g(x)$ has constant amplitude). I could not provide more details for that heuristic.

I've now a further hypothese, which describes a power series solution for $d(x)$ which seems to work perfectly, but again besides the heuristical finding have no proof so far; while I've developed that hypothese in a fairly often applied framework of alternating series, a formal explanation for the correctness of the "mechanic" behind is out of reach for me.

SO is it better to present that power series -hypothese for $d(x)$ as a new question for help for a formal proof, or is it better just to include that hypothese into my previous answer/comment?