The Constants Inside the Fine Structure Constant

On 24 September 2018, mathematician Michael Atiyah announced that he believed that he had cracked the Riemann Hypothesis at the Heidelberg Laureate Forum, which if his proof holds, would represent a very big deal. Not just because the proof would come with a one million dollar price from solving one of the Clay Mathematical Institute’s millennium problems, or because it describes the distribution of prime numbers, but also because it would automatically prove a lot of other mathematical hypotheses that rely on the Riemann hypothesis being valid for their contentions to hold.

For more information about what the Riemann Hypothesis is, we recommend viewing either Numberphile’s 17-minute video or 3Blue1Brown’s 22-minute long video on the topic, both of which are well done but offer different strengths in presentation. What we found interesting in Atiyah’s announcement is that he claims the proof came about because of work he was doing (leaked here?) to analytically derive the fine structure constant from physics, which is fascinating in and of itself.

In the following video from the University of Nottingham’s Sixty Symbols project, Laurence Eaves provides a blissfully short 5-minute long explanation of the significance of the fine structure constant, which is the measure of the strength of the electromagnetic force governing how electrically-charged elementary particles interact.

The three fundamental constants that are combined in the fine structure constant are:

• The elementary charge of an electron (e)
• Planck’s constant, divided by both 2 and pi (h)
• The speed of light (c)

If we include pi, there are technically four universal constants in the formulation. It’s estimated to be very nearly equal to 1 divided by 137, where if it were exactly equal to that ratio, would really be extraordinary because the value 137 happens to also be a Pythagorean prime number. As it happens, the denominator in the fine structure constant is currently estimated to be 137.035999138…, so it’s close, but not quite.