I looked at Jiang's monograph for a little while last night. Here is what I could get from it (I am now quoting from memory, so my terminology and notation may not be exactly the same). If $F$ is a field (of "numbers"), then the field $\overline{F}$ of "isodual numbers" has the same underlying set and addition operation, but multiplication is replaced by the operation $x \ \overline{\bullet}\ y := - (xy)$. The new multiplicative identity is $-1$.
This is mathematically valid, of course: i.e., $\overline{F}$ really is a field. Moreover it is isomorphic to $F$ via the map $x \mapsto -x$, although I couldn't find a clear statement of that. (But somewhat later on I saw references to the isotopy $F \rightarrow \overline{F}$.) Physically speaking, the isodual numbers are supposed to bear the same relation to the ordinary numbers as antimatter does to matter. (I don't know what that means, but I am not a physicist and so am not even going to worry about it.)
Jiang defines a new function $J_2(\omega)$, which is supposed to be some sort of repaired version of the Riemann zeta function. In one of his published works, he claims that the Riemann hypothesis is false -- in fact, he says, the zeta function has no zeros in the critical strip. [Logically speaking, wouldn't that make the Riemann Hypothesis true? Never mind.] From this definition, he immediately deduces proofs of Goldbach, twin primes, primes of the form $n^2+1$, and several other outstanding number theoretic conjectures -- literally immediately, in that I could find no argumentation for them. First these results are stated for "isonumbers" but later on they are stated for the usual integers.
That's about as far as I got. I also noticed, though, that many of the results described in this monograph were first published as papers by the journal Algebra, Groups and Geometries (founding editor: R.M. Santilli). These papers appear on MathSciNet but are not (going to be) reviewed.