I think Mochizuki himself makes a good exposition of the highly spectulative relation (not much more so that the F_1, absolute geometry, noncommutative geometry, etc approaches, if you ask me) between IUT and the Riemann hypothesis. I'll copy the relevant paragraphs from IUT IV here, for future reference (pages 47 to 52):
(i) [...] In particular, the bound under consideration may be written in the form
$$\frac{1}{6}\cdot h \leq \delta + * \cdot \delta^{1/2} \cdot \log (\delta)$$
where "$*$" is to be understood as denoting a fixed positive real number; we observe that the ratio $h/\delta$ is always a positive real number which is bounded below by the definition of $h$ and $\delta$ and bounded above precisely as a consequence of the bound consideration. In this context, it is of interest to observe that the form of the "$\epsilon$ term" $\delta^{1/2}\cdot \log (\delta)$ is
strongly reminiscent of the well-known interpretations of the Riemann hypothesis in terms of the asymptotic behaviour of the function defined by considering the number of prime numbers below a given number.
Indeed, from the point of view of weight [cf. also the discussion of Remark 2.2.2 below], it is natural to regard the [logarithmic] height of a line bundle as an object that has the same weight as
single Tate twist, or, from a more classical point of view, "$2\pi i$" raised to the power $1$. On the other hand, again from the point of view of weights, the variable "s" of the Riemann zeta function $\zeta (s)$ may be thought of as corresponding precisely to the
number of Tate twists under consideration, so a single Tate twist correspondes to "$s=1$". Thus, from this point of view, "$s=\frac{1}{2}$", i.e., the critical line that appears in the
Riemann hypothesis, corresponds precisely to the square roots of the [logarithmic] heights under consideration, i.e., to $h^{1/2}$, $\delta^{1/2}$. [...]
(ii) In [vFr], $2, it is conjectured that, in the notation of the discussion of (i),
$$\lim \sup \frac{\log \left(\frac{1}{6}\cdot h - \delta\right)}{\log(h)}=\frac{1}{2}$$
and observed that the "$\frac{1}{2}$" that appears here is strongly reminiscent of the "$\frac{1}{2}$" that appears in the Riemann hypothesis. [...]
(iii) In the well-known classical theory of the Riemann zeta function, the Riemann zeta function is closely related to the
theta function, i.e., by means of the Mellin transform. In light of the central role played by the theta functions in the theory of the present series of papers, it is tempting to hope, especially in the context of the observations of (i), (ii), that perhaps some extensions of the theory of the present series of papers
- i.e., some sort of "inter-universal Mellin transform" - may be obtained that allows one to relate the theory of the present series of papers to the Riemann zeta function.
(iv) In the context of the discussion of (iii), it is of interest to recall that, relative to the analogy between number fields and one-dimensional function fields over finite fields, the theory of the present series of papers may be tought of as being analogous to the theory surrounding the derivative of a lifting of the Frobenious morphism [cf. the discussion of [IUTchI], $I4; [IUTchIII], Remark 3.12.4]. On the other hand, the analogue of the Riemann hypothesis for one-dimensional function fields over finite fields may be proven by considering the elementary geometry of the [graph of the] Frobenius morphism. This state of affairs suggest that perhaps some sort of "integral" of the theory of the present series of papers could shed light on the Riemann hypothesis in the case of number fields.
(v) One way to summarize the point of view discussed in (i), (ii), and (iii) is as follows: The asymptotic behaviour discussed in (i) suggests that perhaps one should expect that the inequality constituted by well-known interpretations of the Riemann hypothesis in terms of the asymptotic behaviour of the function defined by considering the number of prime numbers below a given number may be obtained as some sort of "restriction"
$$\text{(ABC inequality)|}_\text{canonical number}$$
of some sort of "ABC inequality" [i.e., some sort of bound of the sort obtained in Corollary 2.2, (ii)] to some sort of "canonical number" [i.e., where the term "number" is to be understood as referring to an abc sum]. Here, the descriptive "canonical" is to be understood as expressing the idea that one is not so much interested in considering a fixed explicit "number/abc sum", but rather some sort of suitable abstraction of the sort of sequence of numbers/abc sums that gives rise to the lim sup value of "$\frac{1}{2}$" discussed in (ii). Of course, it is by no means clear precisely how such an "abstraction" should be formulated, but the idea is that it should represent
some sort of average over all possible addition operations
in the number field [in this case $\mathbb{Q}$] under consideration or [perhaps equivalently]
some sort of "arithmetic measure or distribution" constituted by
such a collection of all possible addition operations that somehow
amounts to a sort of arithmetic analogue of the measure that gives
rise to the classical Mellin transform
[i.e., that appears in the discussion of (iii)].
[vFr] M. van Frankenhuijsen, About the ABC conjecture and an alternative (2012)
