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 N is the value of the largest denominator in the “vanilla” Harmonic Series (S_{n} ) and
in the Harmonic Series of Primes (S_{p}). And. of course, assuming N goes to ∞
S_{n} = ln(N)
S_{n} – ln(ln(N)) = ln(N) – ln(ln(N))
S_{n} – S_{p} = ln(N) – ln(ln(N))
S_{n} – S_{p} = ln( N / ln(N) )
S_{n} – S_{p} = ln(π(N))
Now we show the above with comments on the right.
S_{n} = ln(N)

S_{n} is the “vanilla” Harmonic Series 
S_{n} – ln(ln(N)) = ln(N) – ln(ln(N))

let’s subtract ln(ln(N)) from each side of = 
S_{n} – S_{p} = 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). 
S_{n} – S_{p} = ln( N / ln(N) )

Doing some log arithmetic, rewrite
ln(N) – ln(ln(N)) as its equivalent of ln( N / ln(N) ) 
S_{n} – S_{p} = 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 S_{n} – S_{p } over N is the ln of the number of primes over N.
And note that S_{n} – S_{p is, to coin a phrase,} the “Harmonic Series of Composites“ S_{c} = 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!
S_{c} = 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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