>>11605803What you seem to propose is to enumerate all possible proofs. E.g. enumerating all strings and, as a middle-step, checking if they are actually proofs and then checking of what they are proves.
Then every provable statement would eventually get proven and every disprovable statement would eventually get disproven. One issue is that via Gödel, there's statements for which there's neither a proof of it, nor of its negation.
Moreover, this way before proving "1+2=3", you may end up proving "1+2+5^2=3+25", "1+2+6^2=3+36", "1+2+7^2=3+49", and all that crap.
In any case, that's a bit more meta than what Pi_0 is, however, which is more tangible:
A statement S being Pi_0 means, roughly, that it is itself of the form, "forall n·· A(n··)," i.e. the statement S is the claim that a first-order arithmetic statement A holds about all the numbers.
So to disprove the statement, you may just find a counter-example, i.e. an m such that an _arithmetical claim_ A(m) fails. I.e. to disprove is possible in Peano arithmetic, which is nice.
The Riemann hypothesis being Pi_0 means that you can find a counter-example while enumerating over natural numbers like that - which is not apriori obvious (to me anyway) because it's usually formulated about a statement about some complex values function on subsets of C. I.e. it naively sounds like you'd have to find possibly transcendental numbers c satisfying f(c)=0 exactly.
Btw. what you want to get at can in some sense also be done in the logic, and is called Skolemization.
https://en.wikipedia.org/wiki/Skolem_normal_form>>11605789>Has there been any attempts to generalize the zeta function?Of course, in many ways. After all, it's over 150 years old.
Here's a long list of zeta functions, many of which have Riemann's as special case.
https://en.wikipedia.org/wiki/List_of_zeta_functionsA pretty one is an algebraic approach to reason about "primes" in a broader sense, see
https://en.wikipedia.org/wiki/Dedekind_zeta_function 
