It Begs The Question
Hey…I'm just saying… And while we're at it, why are you defending them?

Math Mystery Wrap Up December 13, 2017

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



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