### Sunday, April 16, 2006

## Meissel's Formula, who knew?

It's one of those funny things that years ago I was confronted with the reality that clearly Meissel was using a variant of my prime counting function:

http://mathworld.wolfram.com/MeisselsFormula.html

And, one mathematician to whom I sent it, years ago, said, something like, oh that's just a variant of Meissel's Formula.

He was wrong, but I'll get back to that.

So when posters kept going on and on claiming what I had was Legendre's Method, I could look to see if anyone corrected them to say, no, it's just Meissel's.

No one did.

My prime counting research more than any other research of mine has helped me to see how often posters lied, how routinely other people let them, and how easily people on the newsgroup could be fooled about basic things in the area of prime numbers.

Ok, so there is my prime counting function in its sieve form with y>=sqrt(x) plainly visible in part of Meissel's Formula, so did I copy from there?

Nope. Can't be as I derived my prime counting function from scratch, as a pure thinking exercise.

I just went at it for a couple of weeks thinking about how to count primes, and figured it out, completely off the top of my head.

But more importantly, I emphasize that it is the SIEVE form of my prime counting function with y>=sqrt(x) because that's a child variant, from which you cannot get the parent expression!

The parent expression is a partial difference equation and not a sieve.

But here is where things get really interesting, as there is just no way in hell that short and succinct formula I found is not important, and even if it were somehow not new, there's no reason not to show a short variant of Meissel's Formula, if it were that, which takes up only four lines to display.

But you see, the trouble for mathematicians is that it's not a variant, it's a parent expression, so it blows the lid off the entire problem, so they ignore it, and posters lie about it, screwing up details.

Why is it so important that I keep emphasizing "partial difference equation"?

Because a partial difference equation is just the discrete form of a partial differential equation!!!

So to get the prime count with my prime counting function, I'm doing a discrete integration.

The parallel is an integration of a partial differential equation.

However, I don't do a straight summation, but force the partial difference equation to count slightly off from what it would on its own with the rule that if y>=sqrt(x)

p(x,y) = p(x,sqrt(x))

which explains why the prime distribution is off from the continuous functions.

You see, it's off from the discrete summation as well!

So you have the answer for the why of the prime number theorem, plus the potential to directly calculate the error term, and check the Riemann Hypothesis.

All with one succinct expression of the prime counting function.

So for years now mathematicians have had the capacity to rigorously check the Riemann Hypothesis, and simply chosen not to do it.

They don't really care what the answer is, I think, as I think the answer is that it's false, so if anyone is checking, they aren't talking about what they found.

And I think they aren't checking.

Toss your books on the subject. Math society doesn't really give a damn.

[A reply to someone who asked James what did he mean by “sqrt”.]

Yeah, and I noted that part of the definition is the use of natural numbers.

You people look stupid.

On every point I refute you and you just come back and reply, reply, reply knowing that eventually I'll get tired of it and move on, but you still look stupid.

Trouble is, that your repetitive behavior does work.

People believe you. They trust you. So when I finally wander off, and you people start claiming total victory, they think I've been refuted, when every time I crush your claims, show your objections to be specious, and often show that you don't know much of the subject at hand.

So that happens with Peters and natural numbers, and with this Santos guy and partial difference equations.

The reason none of this matters though is that people naturally expect top mathematicians to step in for a major result and acknowledge its value.

While the people considered to be tops in the field sit back, and hold their breath, hoping that you people ignore the mathematical truth, this charade goes on.

So why do they sit back?

Think it through. ALL of my research kills sacred cows—pet ideas and methods of mathematicians all over the world.

Do you think they want a simple way to check the Riemann Hypothesis?

How do you get grants with that?

How do you pull in millions of dollars for your department with a simple answer?

How do you put your grads through? How do you hand out perks and approval for some while putting out others?

How do you maintain your status if some guy puts out simple answers to big questions?

These people don't think it's that important.

If it were a cure for cancer or appeared to be about life and death, then maybe they would step up, but they can tell themselves that it's JUST about prime numbers.

Get it yet?

Prime numbers aren't important enough to them to tell the truth.

Not when their careers are involved, which is why I get to argue with senseless and often stupid people on newsgroups despite having huge accomplishments.

And yes, if you had made those accomplishments instead of me, they would do it to you as well.

http://mathworld.wolfram.com/MeisselsFormula.html

And, one mathematician to whom I sent it, years ago, said, something like, oh that's just a variant of Meissel's Formula.

He was wrong, but I'll get back to that.

So when posters kept going on and on claiming what I had was Legendre's Method, I could look to see if anyone corrected them to say, no, it's just Meissel's.

No one did.

My prime counting research more than any other research of mine has helped me to see how often posters lied, how routinely other people let them, and how easily people on the newsgroup could be fooled about basic things in the area of prime numbers.

Ok, so there is my prime counting function in its sieve form with y>=sqrt(x) plainly visible in part of Meissel's Formula, so did I copy from there?

Nope. Can't be as I derived my prime counting function from scratch, as a pure thinking exercise.

I just went at it for a couple of weeks thinking about how to count primes, and figured it out, completely off the top of my head.

But more importantly, I emphasize that it is the SIEVE form of my prime counting function with y>=sqrt(x) because that's a child variant, from which you cannot get the parent expression!

The parent expression is a partial difference equation and not a sieve.

But here is where things get really interesting, as there is just no way in hell that short and succinct formula I found is not important, and even if it were somehow not new, there's no reason not to show a short variant of Meissel's Formula, if it were that, which takes up only four lines to display.

But you see, the trouble for mathematicians is that it's not a variant, it's a parent expression, so it blows the lid off the entire problem, so they ignore it, and posters lie about it, screwing up details.

Why is it so important that I keep emphasizing "partial difference equation"?

Because a partial difference equation is just the discrete form of a partial differential equation!!!

So to get the prime count with my prime counting function, I'm doing a discrete integration.

The parallel is an integration of a partial differential equation.

However, I don't do a straight summation, but force the partial difference equation to count slightly off from what it would on its own with the rule that if y>=sqrt(x)

p(x,y) = p(x,sqrt(x))

which explains why the prime distribution is off from the continuous functions.

You see, it's off from the discrete summation as well!

So you have the answer for the why of the prime number theorem, plus the potential to directly calculate the error term, and check the Riemann Hypothesis.

All with one succinct expression of the prime counting function.

So for years now mathematicians have had the capacity to rigorously check the Riemann Hypothesis, and simply chosen not to do it.

They don't really care what the answer is, I think, as I think the answer is that it's false, so if anyone is checking, they aren't talking about what they found.

And I think they aren't checking.

Toss your books on the subject. Math society doesn't really give a damn.

[A reply to someone who asked James what did he mean by “sqrt”.]

Yeah, and I noted that part of the definition is the use of natural numbers.

You people look stupid.

On every point I refute you and you just come back and reply, reply, reply knowing that eventually I'll get tired of it and move on, but you still look stupid.

Trouble is, that your repetitive behavior does work.

People believe you. They trust you. So when I finally wander off, and you people start claiming total victory, they think I've been refuted, when every time I crush your claims, show your objections to be specious, and often show that you don't know much of the subject at hand.

So that happens with Peters and natural numbers, and with this Santos guy and partial difference equations.

The reason none of this matters though is that people naturally expect top mathematicians to step in for a major result and acknowledge its value.

While the people considered to be tops in the field sit back, and hold their breath, hoping that you people ignore the mathematical truth, this charade goes on.

So why do they sit back?

Think it through. ALL of my research kills sacred cows—pet ideas and methods of mathematicians all over the world.

Do you think they want a simple way to check the Riemann Hypothesis?

How do you get grants with that?

How do you pull in millions of dollars for your department with a simple answer?

How do you put your grads through? How do you hand out perks and approval for some while putting out others?

How do you maintain your status if some guy puts out simple answers to big questions?

These people don't think it's that important.

If it were a cure for cancer or appeared to be about life and death, then maybe they would step up, but they can tell themselves that it's JUST about prime numbers.

Get it yet?

Prime numbers aren't important enough to them to tell the truth.

Not when their careers are involved, which is why I get to argue with senseless and often stupid people on newsgroups despite having huge accomplishments.

And yes, if you had made those accomplishments instead of me, they would do it to you as well.