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[4] Inter-universal Teichmuller Theory IV: Log-volume Computations and Set-theoretic Foundations. PDF NEW !! (2020-04-22)
P48
Remark 2.2.1.
In this context, it is of interest to observe that the form of the
“ term” δ1/2 ・log(δ) is strongly reminiscent of well-known interpretations of the
Riemann hypothesis in terms of the asymptotic behavior of the function defined
by considering the number of prime numbers less than a given natural number.
Indeed, from the point of view of weights [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 a single Tate twist, or, from a more classical point of
view, “2πi” 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 ζ(s) may be thought of
as corresponding precisely to the number of Tate twists under consideration, so a
single Tate twist corresponds to “s = 1”. Thus, from this point of view, “s = 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 h1/2,δ1/2.
P49
－ 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.
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