# Can one expect the existence of a relevant approach for a proof of the Riemann hypothesis using Mochizuki's theory? [closed]

Next month at Oxford university, there will have the first workshop outside Asia on the Inter-Universal Teichmuller theory of Shinichi Mochizuki: http://www.claymath.org/events/iut-theory-shinichi-mochizuki

See this recent paper in Nature: The biggest mystery in mathematics: Shinichi Mochizuki and the impenetrable proof, 07 October 2015.

See also the mathoverflow post: Philosophy behind Mochizuki's work on the ABC conjecture

I wanted to post the question in title but I've found an equivalent question posted on Quora 11h ago (without answer), and I think it has its place on mathoverflow, so I just reproduce it below:

Mochizuki has published a series of 6 papers (as of 11/11/2015) on Inter-Universal Teichmüller Theory on his webpage. In the one which contains the proof of the abc conjecture (4/6), he relates his methods also to the Riemann hypothesis (pages 47-53). Can someone explain how strong this link is? Is there a way to estimate - given the information in this paper - how likely it is that the Riemann hypothesis can be proved using Mochizukis methods?

The main reason why I ask this question today is that one author of the abc conjecture is visiting my institute for 4 months, I've talked with him several times this week, and I discovered by him the polemic around Mochizuki's proof. Moreover because there will have this famous workshop next month, I think it would be a good opportunity to clarify this question also.

## closed as primarily opinion-based by user9072, Lucia, Alexey Ustinov, Noah Snyder, Ben LinowitzNov 15 '15 at 16:19

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.

• I'm sorry but this is a terrible question. What is the point of speculating on how Mochizuki's work (which no one seems to have seriously evaluated), applies or does not apply to RH? – Lucia Nov 15 '15 at 14:45
• The phrase "published a series of 6 papers...on his webpage" significantly lowers the bar for what it means to "publish". This highly speculative question has no place on MO. – nfdc23 Nov 15 '15 at 15:39
• What motivates you to ask? Is this something you think will be useful for your own research, or is it sort of idle curiosity? – Todd Trimble Nov 15 '15 at 16:05
• @ToddTrimble: The main reason why I ask this question today is that one author of the abc conjecture is visiting my institute for 4 months, I've talked with him several times this week, and I discovered by him the polemic around Mochizuki's proof. Moreover because there will have this famous workshop next month, I think it would be a good opportunity to clarify this question also. – Sebastien Palcoux Nov 15 '15 at 16:44
• I think it's fair to say that it has been difficult for most people to get into Mochizuki's work in a serious way and so any discussion on whether it is correct and what the implications to other questions it might have are just premature. Hopefully, next month's workshop will help change this. I think the best thing to do is wait a month. – Felipe Voloch Nov 15 '15 at 17:44

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)