Thank you for this. Is that a typo - does one replace $p_0$ in the infimum by just $0$ (which I believe then simply follows from Markov's inequality)?

The book by Hall was very lovely to me springer.com/gp/book/9781461471158 I think it has a particularly great explanation of spin, which is often quite confusing. That said, I think it's usually best to mix in a few different references to get a holistic perspective. Griffith's QM and "QFT for the gifted amateur" were good for skimming overviews for me.

I personally think that working with motives is a little out of reach at the moment. I think arguably the three most important instances of cohomology theories (conjecturally) unified by motives are Betti, de Rham and etale cohomologies. But I think the first two are kind of indistinguishable on simplicial complexes, where TDA happens. The last one becomes probably even weirder, because I'm not sure if there's a meaningful notion of schemes in computational / approximate settings.

Thanks for the pointer, I simply didn't know about that formula (I'm aware of such a general expansion, but didn't know that it would work for $\zeta$). Is there a similar formula for multiple zeta functions?

Great answer! In the case of Hodge conjecture, how does one obtain the description in terms of functors, starting from the usual formulation using rational (p,p)-forms?

Thanks! Those two theorems resolve my concerns mostly. I'll leave the question unanswered for a bit to see if others have more to say about the matter.

Thanks. I'll read the reference, but do you know whether the Littlewood's theorem translates to an impossibility of a power bound of $|\pi(x) - Li(x)| = O(x^\alpha)$ with $\alpha \leq \frac12$?