# 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.

Epilogue – Chronicle of Composites Project

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   –   Sp   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