In the version that there are recursively enumerable sets that are not recursive, that seems fair. Dieudonné certainly once said that if certain problems are not algorithmically soluble, then we should care more about other things. (But I disagree with the tenor of the question. If 0 = 1 results from some high-powered proof, that shifts the foundational debate back to a century ago. But some illumination will come out of it, as axiomatic set theory came out of the paradoxes.)

Should really add that the original theory is really not known to work (after 45 years). There are ways round this (absolute Hodge cycles, motivic cohomology), but these are less accessible.

Your question is related to the Gamma function (en.wikipedia.org/wiki/Gamma_function) at $-1$; but the product is meaningless, the Gamma function has a singularity there, and this all has been known for two centuries.

This is about the sign of the second derivative.

I'm thinking about the Frobenius for the curve being $\pi$, or in other words the Hecke character associated to the curve by the theory of complex multiplication. But my explanation was hurried, and I may need to edit it.

Quite hard to sharpen anything of Hooley's.

Plenty of analysis. Starting from a computer science background, you should probably concentrate on developing some feeling for inequalities.