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An Actual Math Mystery November 12, 2017

The Mystery Formula

 

On Nov. 1 I found the following sheet of notebook paper with some formulas written on it. 

It was tucked away in a remote corner of a bookshelf.

 

Here it is transcribed with some annotation and easier to read:

 

ln(Cp) + Sp = Sn

next  convert  Cp         see  https://en.wikipedia.org/wiki/Prime-counting_function

ln( n / ln(n) ) + Sp = Sn

ln(n) – ln(ln(n)) + Sp = Sn

Sp = Sn+ ln(ln(n)) – ln(n)

ln(n) is same as Sn so make it so below

Sp = Sn + ln(ln(n)) – Sn

Sp = ln(ln(n))

 

The above was obviously written by me and it was also obvious (to me) that it concerned the relationship of the “vanilla” Harmonic Series and the Harmonic Series of Primes. As a refresher for the reader…

The “vanilla” Harmonic Series is:

The Harmonic Series of Primes is like the “vanilla” Harmonic Series except that all of the denominators of the terms are only primes:

The paper with the Mystery Formulas must have been part of an earlier project that I had abandoned about 3-4 months ago when I was not making any progress. Anyway, that project concerned trying to find a simple “proof” of the divergence of the Harmonic Series of Primes.

 

As I said, I had eventually abandoned the “simple proof” project some months ago because I was not making progress. But what is VERY STRANGE is that:

What is  written on that sheet of notebook paper above constitutes a “simple proof” I had been looking for!… Sort Of… Let me explain… It’s fascinating.

 

But first, I need to explain the nomenclature used. That is, what the “symbols” represent (it’s extremely simple but needs to be understood).

Sp is the sum of the terms of the Harmonic Series of Primes. That is, Sp  is the “value” of the series.

Sn is the sum of the terms of the “vanilla” Harmonic Series

Cp is the Count of the prime numbers in either Sp or Sn

Ln is the natural log

 

 

That said, the last formula on the sheet of notebook paper (above) is:

Sp = ln(ln(n))

The above formula expresses the “value” of the Harmonic Series of Primes.  You’ll find the above formula that expresses the value of  Sp     in many places (e.g. Wikipedia).  Anyway, as n goes to infinity we can easily see that Sp diverges!  That is,  the Harmonic Series of Primes diverges.

Sp diverges! This is what I was trying to show, in a simple manner, in the project I abandoned months ago (due to lack of progress).

 

 

So… Do the simple formulas on the Mysterious Sheet of Notebook Paper

actually represent a simple proof that I had been searching for?

Wellllllllll….. maybe.

It totally depends on whether the first formula is true. And that formula is:

Formula 1  –>       ln(Cp) + Sp = Sn

 

So is it true? Although I don’t remember doing so, I had apparently written this stuff down some months ago. If I had seen it somewhere I am pretty sure I’d remember. Now what’s really interesting is that, when I look at that first formula (just above),

I can’t think of any reason why adding the log of the count of terms with a prime denominator, to the value of the Harmonic Series of Primes, would equal the value of the vanilla Harmonic Series!!!!??

I can’t think of a reason; can you? But apparently I thought that in the past! Anyway, I’ll tell you what I can do. I can “test” the formula over billions of numbers to see if it looks like it holds water. So, while that’s not a proof, per se, it would be good evidence one way or the other. Luckily, doing such an experimental test would be easy because I had already written a SumOfReciprocals” program to help with my first attempt at finding a simple proof of the divergence of the Harmonic Series of Primes.

All that was needed was a few lines of code to display, at the end of each test, the values of each term in Formula 1 (above). For each test execution, the SumOfReciprocals program has already accumulated the values for Cp, Sp, and Sn so all that was needed was to display them in the log. Just below you can find a screen print of one such SumOfReciprocals test execution.  Click on it to see it full size.

As it turned out, the test results appear to show that the left side of the equation asymptotically approaches the right side of the equation. i.e.

ln(Cp) + Sp ~ Sn

Here is a spreadsheet showing the results of the above where the values of the denominators range from 1..109 (1 to 1 Billion).

 

 

The spreadsheet and associated chart show the results of

 

 

Sn

_____________________________

ln(Cp) + Sp

 

 

As the above ratio approaches 1, it demonstrates that the left side of the equation approaches the right side. That is, that ln(Cp) + Sp approaches Sn

So where does that leave us?

-1-  Based on some experimental evidence, it’s my conjecture that      ln(Cp) + Sp ~ Sn

-2- If the conjecture is true, then the series of formulas on the notebook paper constitutes a very simple proof of the divergence of the Harmonic Series of Primes.

 

So, all that said, how would we prove     ln(Cp) + Sp ~ Sn ?     I’ve thought about it for some time and I can’t think of any reason why ln(Cp) would play any role at all! But the experimental results appear to powerfully indicate otherwise!

It’s all a mystery and now the original project has morphed into something much more interesting…

Proving (or disproving)

ln(Cp) + Sp ~ Sn

 

 

The End

 

 

 

 

 

 

 

 

 

 

 

 


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