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

Epilogue To The Composite Chronicle December 20, 2017

Math Mystery Epilogue

 

This post is a kind of “stream of consciousness” epilogue to the Math Mystery Project; aka the Harmonic Series of Composites project. It’s a “story” of sorts; an additional chronicling of the “ending” of the project. Anyway, none of what follows will make any sense at all until/unless you’ve read the first 3 “parts” of the “Math Mystery” story. Here’s the links if you want to get started.

1 https://itbegsthequestion.com/ an-actual-math-mystery/

2 https://itbegsthequestion.com/ math-mystery-followup/

3 https://itbegsthequestion.com/ math-mystery-wrap-up/

Ok, we’re back. As you probably noticed, this “project” ended up being something completely different than what it started out as. That’s not uncommon either with the projects or with life in general (duh). We start out investigating something, notice something “odd” or interesting, or shiny, and then it’s off in another direction; sometimes a bunny trail into the weeds but sometimes something meaningful and fascinating and worth remembering.

Anyway, at the end of Part 3 of the Math Mystery, I decided to find and email some professional mathematicians to see what they thought of what was “discovered.” I’ve not heard back from the first 2 yet but here is the email I sent to them (btw I am including this “first” email and parts of some others and the responses in order to document what was being thought at the time…. a personal historical log. Ooh! A “phlog!” Anyway, here’s the first email to two pros:

Dear Mr. Roelandts,

I am hoping you can solve a problem for me (or direct me to a doc that will).

π(n) is the count of primes in 1..n (the prime counting function)
let Sp be the Harmonic Series of Primes thru n
(1/1 + 1/2 + 1/3 + 1/5 + … 1/p)
let Sn be the Harmonic Series thru n
(1/1 + 1/2 + 1/3 + 1/4 + 1/5 + … 1/n)

Then it’s my conjecture that:

ln(π(n)) + Sp ~ Sn

In words:

The log of the number of primes
plus
the sum of terms for the Harmonic Series of Primes
asymptotically approaches
the sum of terms for the “vanilla” Harmonic Series
If we manipulate the above formula we will get
Sp ~ ln(ln(n))
which we see a lot.
 —
In addition, if we actually perform the calculations
(at least thru n = 1 billion using doubles in a C# program)
the formula appears to hold water… but why? Why should π(n) be part of
the relationship between Sp and Sn? Why should π(n) be part of the relationship
between the Harmonic Series and the Harmonic Series of Primes?
 —-
Also see my blog article about the above at
Be advised that I am a rank amateur (capital R) but try as I might, I can’t find any articles or papers about this. What am I missing (if anything)?
Thanks in advance,

 

 

I also copied the above email to a friend of mine in California (JGM) and here is what transpired:

Gerry – I sent your email to a mathematics friend and this is what he said

looks like a number theory cleverness of Ramanujan (encouraged by Hardy)”

Note… emphasis is mine above.

To which I responded

I don’t know what that means. Anyway, I can see WHAT the
answer is both experimentally via computer and by doing the
math (algebra 1). It’s way way simple. Nothing
clever going on. I’m too much of an amateur to be clever. It’s
just that I can’t fathom WHY the log of the count of the primes
in the series would/should be related to the series “value” (the
sum). And I can’t fathom WHY there seems to be no significant
associated literature that I can find about this series (the
Harmonic Series of Composites aka Sum of Reciprocals of
Composites). Is Google trying to hide the answer from me? Uh
oh… I’m starting to connect conspiracy dots that don’t exist!!!
WAIT! What’s that noise!? OH NO!! There’s a black
escalade outside!! Gotta go. Just enough time to click on
SEND.
gerry

 

Then I googled “cleverness of Ramanujan” as mentioned above and found out what they were (presumably) talking about so I wrote back to JGM.

John,
Thanks to google I now get the “cleverness of Ramanujan” reference which I’ll take as a compliment though (largely?) maybe undeserved. Here is a link
https://www.wired.com/2016/04/who-was-ramanujan/

The wired magazine article linked to just above contains the following (emphasis below is mine).

Again, they began unpromisingly, with rather vague statements about having a method to count the number of primes up to a given size. But by page 3, there were definite formulas for sums and integrals and things. Some of them looked at least from a distance like the kinds of things that were, for example, in Hardy’s papers. But some were definitely more exotic. Their general texture, though, was typical of these types of math formulas. But many of the actual formulas were quite surprising—often claiming that things one wouldn’t expect to be related at all were actually mathematically equal.

btw, the Wired Magazine article referenced above was written by Stephen Wolfram. You may already know about him but if not, just google him (and prepare to be amazed).

btwbtw with regard to “often claiming that things one wouldn’t expect to be related at all were actually mathematically equal”  ……. It has VERY often been my experience that what often shines a bright light on relationships of “number stuff” are the results of actually doing computations… BILLIONS and BILLIONS of computations (we need to write programs for this!). Then analyzing and comparing the results (the numbers) using things like logs (which often plays a role in number theory) and using a good scientific calculator ( like the one that comes with Windows, or Google’s (just type “calculator” in the address bar of the Chrome browser)). All this just to help us “see” relationships. Of course, we need to have a feel for what we may want to compare and then we need to actually write the code to compute that stuff. Then the results will hint that we need some additional result which we dutifully add to the code… rinse and repeat. Of course, not every number computed will yield interesting results so we need to be prepared to “waste” many hours of effort writing and revising these programs. And keep in mind… we don’t get paid for this shit.

btwbtwbtw… the mathematicians of the past would very often do the same thing (perform gazillions of computations) and for the same reasons; but they did it by hand!… when paper was not easy to come by! Can you imagine what THEY would have thought, and given, to have a computer!?

Time to go… the black escalade is back.

gerry

So the above email chain is what I was thinking at the time. Then about a week later while googleing “harmonic series of composites” and related terms, I ended up on the OEIS web site and finding its founder, Neil Sloane.

IMPORTANT! Follow these links
to learn about OEIS and Neil Sloane:

So who’s Neil? And what is OEIS?

Science News https://www.sciencenews.org/article/pattern-collector

Wired Magazine Meet the Guy Who Sorts All the World’s Numbers in His Attic

The Guardian  https://www.theguardian.com/science/alexs-adventures-in-numberland/2014/oct/07/neil-sloane-the-man-who-loved-only-integer-sequences

For OEIS site, see —->> https://oeis.org/

and for my “contribution” to humanity see

https://oeis.org/search?q=A296358&language=english&go=Search

 

So I emailed Mr. Sloane to see what a recognized expert had to say about all this Harmonic Series of Composites stuff.  The email and his response follows:

Dear Mr. Sloane,

I am hoping you can help to solve a (simple?) series related problem for me (or direct me to a document that will). I can’t remember how I arrived at your name as a likely candidate for helping but it (of course) involved numerous google searches that eventually landed me on OEIS and one of its pages:

https://oeis.org/wiki/Talk: Harmonic_series_of_the_ composites

That page did not help directly but with just a few clicks around OEIS I stumbled on you. Anyway, with all that said, here is the short story… i.e. “Just the problem” as Sgt Joe Friday would say.

The two most “famous” series are the “vanilla” Harmonic Series, and of course, the Harmonic Series of Primes. The shorthand/symbols I use for these series is Sn and Sp (S for Series, n for natural numbers, and p for primes). When we go to infinity the

vanilla Harmonic Series ~ Ln(N)
and the Harmonic Series of Primes ~ Ln(Ln(N))
or…
Sn ~ Ln(N)
Sp ~ Ln(Sn)     or,    Sp ~ Ln(Ln(N))

We see the above in a lot of places.

But what about the third leg of the Harmonic Series “Triad?” What about the Harmonic Series of Composites ( we’ll call it Sc )? We find references and extensive literature all over the internet about the vanilla Harmonic Series (Sn ) and about the Harmonic Series of Primes (Sp) but the number of references to the Harmonic Series of Composites is just about nil! Why? Especially when its “value” is so interesting!!! And this brings us to the main points of this whole discussion.

1. What is the value of Sc ?
2. Why?

So what is its value? Of course, when we take all of the primes (p) away from the natural numbers (n) we are left with just composites.
Sc = Sn – Sp

Let’s massage the above a little and we’ll get…

Sn = Ln(N) and Sp = Ln(Ln(N))
so…
Sc = Ln(N) – Ln(Ln(N))

Here is where the magic happens with a little log arithmetic

Sc = Ln( N / Ln(N))
AND Note that … N / Ln(N) is the prime counting function π(N) so…
Sc = Ln(π(N))

or, said in words,
The Harmonic Series of Composites (aka Sc )
approaches the log of the number of primes in N as
N → infinity!

Note that experimental results (computations) support the above at least thru N = 1 billion. You will see this (and more) if you read the related blog articles (links are further down).

With all that said, some questions…  Did I miss something? Did I make some bone-headed mistake along the way?
Why don’t I find any literature about any of this? And, do you know of any literature/documents that cover the Harmonic Series of Composites?
I am Especially looking for an “intuitive” reason for WHY the log of the count of primes should have anything to do with the value of the Harmonic Series of Composites.

Thanks for any insights or other information you may have.

Gerry

P.S. Below are 3 links to a 3-part “story”/ history about all of this on my blog if you are interested. And once again, thanks for any insights you might have on the above.

1 https://itbegsthequestion.com/ an-actual-math-mystery/
2 https://itbegsthequestion.com/ math-mystery-followup/
3 https://itbegsthequestion.com/ math-mystery-wrap-up/

Mr. Sloane replied to the above email with this response (again, emphasis is mine).

Dear Gerry,
I agree with your estimate for Sum_{k=1..n} 1/composite(k).

Although the OEIS has many similar sequences of fractions – here is a list:
Sum 1/n: A001008/A002805, Sum 1/prime(n): A024451/A002110 and A106830/A034386, Sum 1/nonprime(n): A282511/A282512 –

it did not have that one. We did have the successive numerators (A282511)
but not the denominators, so I created A296358 for them (actually the denominators of Sum 1/nonprimes, A282512, are essentially the same sequence). So Sum 1/composite(n) is now A282511/A296358. I then added your interesting comment about the asymptotics to A296358. Although it is easy to prove, it is a nice observation, and I had not seen it before.
Thanks for writing.

Neil Sloane

 

Easy and simple are big pluses in math. And, the man who has seen it all in the world of integer sequences hadn’t seen this one! So it turns out that I was not crazy!! And he agrees with my proof. But… But… he did not hazard a guess as to WHY π(N), or more specifically, Ln(π(N)) should play a role in the value of the Harmonic Series of Composites… But it does… and it is! Once again, it appears to be another case of what was said in the Wired Magazine article about Ramanujan:

things one wouldn’t expect to be related at all were actually mathematically equal.”

So Mr. Sloane created my own entry/page in OEIS. It’s

http://oeis.org/search?q=A296358&language=english&go=Search

As said in the Guardian article

It is also a badge of honour to get a sequence accepted – although Neil and the administrators tend to be very generous in their selection criteria. Still, you need to think up something that hasn’t been thought up before!”

and…

Anyway… The “Mathematical equivalence” discovered during this project was and IS VERY surprising and unexpected.

Sc = Ln(π(N))

And it’s amazingly simple and easy to prove as Mr. Sloane will attest to. And one of the biggest surprises (and mysteries) is that, as far as I can tell, it’s been essentially unknown!… Even to Mr. Sloane! And just as surprising is that it all started with some unexplained notes on a wrinkled piece of notebook paper stuffed into a remote corner of an IKEA bookshelf! Life can indeed be VERY mysterious.

I wanted to remember all this for a long time which is why I spent the hours documenting it above, and in the first 3 articles.

 

The End

 

 

 

 

 

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