Math Mystery Wrap Up

If you haven’t already done it, you will need to read the following 2 articles to make total sense of this one.

A Math Mystery       and        A Math Mystery Follow Up

And after reading this article there’s a 4th and final one

Epilogue – Chronicle of Composites Project

 

Let’s just start with the formula for the “vanilla” Harmonic Series. 

Then massage it in 4 simple steps and see where it takes us!

Note:  in the following    is the value of the largest denominator in the “vanilla” Harmonic Series  (Sn )  and

in the Harmonic Series of Primes  (Sp).  And. of course, assuming  N  goes to 

 

Sn      =       ln(N)

Sn – ln(ln(N))     =       ln(N)  –   ln(ln(N))

Sn   –   Sp        =       ln(N)  –   ln(ln(N))

Sn   –   Sp        =       ln(   N  /   ln(N)  ) 

Sn   –   Sp        =      ln(π(N))


Now we show the above with comments on the right.

Sn      =       ln(N)

 

Sn is the “vanilla” Harmonic Series

 

Sn   –   ln(ln(N))     =       ln(N)  –   ln(ln(N))

 

let’s subtract  ln(ln(N)) from each side of  =
Sn   –   Sp     =       ln(N)  –   ln(ln(N))

 

On left side, rewrite  ln(ln(N)) as its equivalent which is “Harmonic Series of Primes” (hence the subscript p).
Sn   –   Sp     =       ln(   N  /   ln(N)  ) 

 

Doing some log arithmetic, rewrite

ln(N)  –   ln(ln(N))   as its equivalent of

ln(   N  /   ln(N)  ) 

Sn   –   Sp     =      ln(π(N))

 

 

 

(   N  /   ln(N)  )    is  really the “Prime Counting Function” so if we take the ln of it  we  get

ln(π(N))  which is another way of saying that  Sn   –   S over N  is  the ln of the number of primes over N.

 

And note that Sn   –   Sp    is, to coin a phrase,

the “Harmonic Series of Composites

Sc  =  ln(π(N))

 

So that’s it.  In just a few simple steps we are able to show that the “Harmonic Series of Composites” (to coin a phrase)  is simply the log of the Prime Counting Function!   

Sc  =        ln(π(N))

 

AND we have seen  that the actual computations (thru N = 1 Billion) supports this!  But we (I) still do NOT have any intuitive feeling as to WHY!

 

AND,  try as I might,  I can NOT find any references to this “discovery” anywhere on the internet.  Oh well… time to accept it for what it is… an interesting mystery.

 

The End