**by far**the most accurate measurement of the fine-structure constant

(measured at the scale of the electron mass) was made by Gerald Gabrielse

and colleagues from Harvard, Cornell and RIKEN.

Measuring the "magnetic moment" of a

*single electron*in a "quantum cyclotron" and

inserting that value into state of the art QED equations, the value they determined is:

[math]\alpha^{-1}=137.035999084(51)[/math],

which means that the fine-structure constant lies somewhere in between:

[math]\alpha^{-1}=137.035999135[/math], and

[math]\alpha^{-1}=137.035999033[/math].

Now, these values were determined back in 2008, and since then,

no significant improvement in accuracy was ever accomplished,

despite enormous improvements in both the design of the equipment

and the QED equations themselves.

Thus, many scientists now suspect that further refinements in

the value of the fine-structure constant may not even be possible,

and that the last two values represent, for all practical purposes,

the actual lower and upper bounds of the fine-structure constant as

measured at the scale of the electron mass. Even Gabrielse himself

believes that the above values will hold for a long, long time to come.

Given the above facts, it now seems that in order to be correct,

**any**mathematical expression which results in the fine-structure constant

must not only match the above experimental value

**, but must also include,**

*exactly*within it's form, some simple way of expressing, with the same degree of absolute accuracy,

those seemingly inherent lower and upper bounds.

Now, some of you may remember this article: http://donblazys.com...l_numbers_3.pdf

which describes a finding that occured right here at Hypography many moons ago.

Well, thanks to a great fellow named Lars Blomberg, (who found it via the OEIS)

we now have values of [math]\varpi(x)[/math] to [math]x=10^{15}[/math].

This information was crucial in not only greatly improving the "counting function",

but also allowed me to derive

*these*values of the fine-structure constant as well:

[math]\alpha^{-1}=137.035999084=(A^{-1}*\pi*e+e)*(\pi^{e}+e^{(\frac{-\pi}{2})})-\frac{1}{(6*\pi^{5}*e^{2}-2*e^{2})}[/math]

[math]\alpha^{-1}=137.035999135=(A^{-1}*\pi*e+e)*(\pi^{e}+e^{(\frac{-\pi}{2})})-\frac{1}{(6*\pi^{5}*e^{2}-2*e^{1})}[/math]

[math]\alpha^{-1}=137.035999033=(A^{-1}*\pi*e+e)*(\pi^{e}+e^{(\frac{-\pi}{2})})-\frac{1}{(\mu*e^{2}-2*e^{\frac{5}{2}})}[/math]

where [math]\mu=1836.15267247(80)[/math] is the "proton to electron mass ratio", and

[math]A=2.566543832171388844467529...[/math], is that very special "Blazys Constant"

which generates all of the prime numbers, in sequential order, by the following simple method:

Note that the whole number part is the first prime [math]2[/math], and that:

[math]((2.566543832171388844467529...)/2-1)^{-1}[/math]

is approximately:[math](3.530176989721365539402422...)[/math],

where the whole number part is the second prime [math]3[/math], and that:

[math]((3.530176989721365539402422...)/3-1)^{-1}[/math]

is approximately [math](5.658487746849688216649061...)[/math],

where the whole number part is the third prime [math]5[/math], and so on.

(In short, we divide the approximate number by it's whole number part, subtract [math]1[/math],

and take the reciprocal of the result to get the next approximate number whose whole number part is the next prime!)

That the fine-structure constant is thus related to the prime numbers was also discovered (independently) by Ke Xiao

who publised his findings in a paper entitled "Dimensionless Constants and Blackbody Radiation Laws" in

The Electronic Journal of Theoretical Physics. (It can be "Googled".)

I will post Lars Blomberg's determinations of [math]\varpi(x)[/math] in my next post,

and a revised "polygonal number counting function" in the post after that.

There's more... a lot more... but that's all for now.

It's good to be back.

Don.