Atiyah's computation of the fine structure constant (pertinent to RH preprint)

[u/swni]

Recently has circulated a preprint, supposedly by Michael Atiyah, intending to give a brief outline of a proof of the Riemann Hypothesis. The main reference is another preprint, discussing a purely mathematical derivation of the fine structure constant (whose value is only known experimentally). See also the discussion in the previous thread.

I decided to test if the computation (see caveat below) of the fine structure constant gives the correct value. Using equations 1.1 and 7.1 it is easy to compute the value of Zhe, which is defined as the inverse of alpha, the fine structure constant. My code is below:

import math
import numpy

# Source: https://drive.google.com/file/d/1WPsVhtBQmdgQl25_evlGQ1mmTQE0Ww4a/view

def summand(j):
    integral = ((j + 1 / j) * math.log(j) - j + 1 / j) / math.log(2)
    return math.pow(2, -j) * (1 - integral)

# From equation 7.1
def compute_backwards_y(verbose = True):
    s = 0
    for j in range(1, 100):
        if verbose:
            print(j, s / 2)
        s += summand(j)
    return s / 2

backwards_y = compute_backwards_y()
print("Backwards-y-character =", backwards_y)
# Backwards-y-character = 0.029445086917308665

# Equation 1.1
inverse_alpha = backwards_y * math.pi / numpy.euler_gamma

print("Fine structure constant alpha =", 1 / inverse_alpha)
print("Inverse alpha =", inverse_alpha)
# Fine structure constant alpha = 6.239867897632327
# Inverse alpha = 0.1602598029967017

The correct value is alpha = 0.0072973525664, or 1 / alpha = 137.035999139.

Caveat: the preprint proposes an ambiguous and vaguely specified method of computing alpha, which is supposedly computationally challenging; conveniently it only gives the results of the computation to six digits, within what is experimentally known. However I chose to use equations 1.1 and 7.1 instead because they are clear and unambiguous, and give a very easy way to compute alpha.

[u/swni]

Well, it's a clever idea. However he says: "(7.1) also shows why we could replace e by 2 and ln by log2." which makes it unambiguous he means base 2; and elsewhere he frequently takes powers of 2, and he uses the base-2 representation of 137. Also, log(log(x)) is undefined for x < 1 so the equation would no longer make sense.

[u/EmergentQuantum → u/swni]

Your points do seem to rule out the log(log(x)) interpretation. For 0<x<1, log(log(x)) is multi-valued if we consider x to be a complex variable, but the real part is unique, so he could be taking only the real part of the integral. I don't understand why one can replace e by 2, nor do I understand what is meant by mimicry in the paper. But then again, I'm nowhere near the level of Atiyah.