Sometimes answers can be kind of just there. Like for a LONG time there was a puzzle over why the count of prime numbers could have ANY connection with continuous functions. But I found an easy answer with a partial differential equation I found that follows from my way to count prime numbers.
My prime counting method leads to a difference equation from which I can get to a partial differential equation:
P'y(x,y) = -(P(x/y,y) - P(y, sqrt(y))) P'(y, sqrt(y))
I talk more about in an earlier post I like for reference, and will admit maybe should discuss it more, given its implications but feel like am primarily repeating myself.
Gist of it though is that there is a sieve form for a prime counting function that counts by calling itself. That allows for a difference equation, which does the same, but is a difference equation as it can tell itself when a number is prime. And that leads to the partial differential equation which is the continuous function.
Here's a post I made in 2009 which shows them all together. I rarely do that for some reason.
So past mathematicians were approximating the integration of the partial differential equation, with no clue. But it's a simple, succinct explanation for how the count of primes could be approximated by a continuous function, which isn't exciting. I like to say, it isn't math sexy.
It's like if Gauss were around I could explain things that he died not knowing, but did he miss much? I'm not sure. Or Euler or even Fermat maybe, oh and definitely Riemann. Getting THE answer, would it have been worth it to them? I like to think, yes. But they died with the prime count connection to continuous functions a big mystery, and not something easily explained.
But no, no dramatic, extraordinary and profound possibly mystical explanation needed for prime counts connected to x/ln x or Li(x) or any continuous function. Just a rather mundane, simple and straightforward mathematical explanation.
In our time the answer doesn't seem to get much excitement going, which is ok with me, though also puzzling. But I know that sometimes answers can be, well just not that exciting. I find that intriguing to some extent, but it's just one thing I have. So um yeah, I explained the primes connecting to continuous functions thing, years ago, and moved on to other things.
There is no other like it though.
And there are clues to how big the partial differential must be.
There are no other known partial differential equations that follow directly from a method to count primes. It comes from a difference equation which is ALSO unique in that there are no others that can count primes.
Yup, worth saying again, the difference equation counts prime numbers, when properly constrained. There is no other that does.
Finds them on its own which rattled me for years. Shifted how I looked at the math.
It is one of my favorite discoveries. Also one of the easiest of the BIG ONES to check I think. And one that gives a LOT of perspective.