Tell me more ×  1) Can the Riemann Hypothesis (RH) be expressed
as a $\Pi_1$ sentence?  More formally,  2) Is there a $\Pi_1$
sentence which is provably equivalent to RH in PA?  (This is
mentioned in P. Odifreddi, "Kreiseliana: about and around George
Kreisel", AK Peters, 1996, on page 257. Feferman mentions that
when Kreisel was trying to "unwind" the non-constructive proof
of Littlewood's theorem, he needed to deal with RH. Littlewood's
proof considers two cases: there is a proof if RH is true and
there is another one if RH is false. But it seems that in the
end, Kreisel used a $\Pi_1$ sentence weaker than RH which was
sufficient for his purpose.)   So we have two proofs that the RH
is equivalent to a $\Pi_1$ sentence.  The first is DMR 1974:
$$\forall n >0 \ . \ \left(\sum_{k \leq \delta(n)}\frac{1}{k} -
\frac{n^2}{2} \right)^2 < 36 n^3 $$  The second is by J.
Lagarias:  $$\forall n>60 \ .\ \sigma(n) < \exp(H_n)\log(H_n)$$
But both use theorems from literature that make it difficult to
judge if they can