JSH: No tables, Merten's and prime politics
jst...@msn.com
to notice that
((p_j - 1)/p)*...*(1/2)
where p_j is the jth prime less than or equal to a natural number x,
logically gives the probability that x is prime.
The logic is simple enough, and it follows just from noting that the
probability that x does not have p as a factor if p is a prime less
than or equal to sqrt(x) is roughly (p-1)/p.
But Gauss guessed that the probability of x being prime was more like
1/(ln x).
A result from Mertens shows that the idea I'm using would give
1.122.../(ln x), differing with Gauss.
And I think that's it. Gauss was considered too big to be wrong, so
mathematicians just came up with convoluted ways to ignore what the
correct answer should be.
The clue is no tables.
I looked and saw tables that show the count of prime numbers with Li(x)
and x/(ln x), but nothing that compared them with the count predicted
by the probabilistic approach.
The simple reply to this post is not to repeat that the entire
mathematical establishment has already considered the probabilistic
approach and found problems with it--as I think they are just
rationalizing--and just give the evidence.
Trot out a table.
I think some of you just are too trusting of a lot of people who claim
something for over a hundred years, as if that means it must be true.
But I come from a physics background where we learned of ideas that had
stood for thousands of years that were wrong. Sure you may say we are
modern today, but we are still the same basic types of people.
Humanity has not had time to evolve much in that time period.
I suggest to you that much of what you think you know about primes and
the prime distribution is slightly off, to keep with what Gauss
believed, and that mathematicians have protected him by simply not
comparing, but hey, I could be wrong, as I often am.
The correct reply is to just trot out some numbers, showing how well
this idea actually compares.
Show the numbers.
James Harris
Jim Lanson
<jst...@msn.com> wrote in message
news:1155788192....@m73g2000cwd.googlegroups.com...
>
> But I come from a physics background where we learned of ideas that had
> stood for thousands of years that were wrong. Sure you may say we are
> modern today, but we are still the same basic types of people.
> Humanity has not had time to evolve much in that time period.
you graduated with a "Slow Physics" degree, that studies only slow things,
and then you were still wrong.
> I suggest to you that much of what you think you know about primes and
> the prime distribution is slightly off, to keep with what Gauss
> believed, and that mathematicians have protected him by simply not
> comparing, but hey, I could be wrong, as I often am.
>
> The correct reply is to just trot out some numbers, showing how well
> this idea actually compares.
>
> Show the numbers.
>
do your own research, and google for it
==>> * crackpot * <<==.
>
> James Harris
>
jshs...@yahoo.com
Alright James, here is a table provided by Tim Peters in another one of
your threads.
---------------------------------------------------------------------------------------------------------------------
Here's a table for you, tabulating values at each 50 million from 50
million
through 10^9.
M: x/10^6
pi: pi(x)
li: li(x), the definite integral from 0 to x of dt/ln(t)
x/ln(x): what it says
jsh: x * product of 1-1/p over all prime p <= sqrt(x)
mertens: 2*e^-gamma * x/ln(x), the approximation to `jsh`
given by Mertens's theorem
pi/li: ratio pi / li
pi/ln: ratio pi / (x/ln(x))
pi/jsh: ratio pi / jsh
M pi li x/ln(x) jsh mertens pi/li pi/ln
pi/jsh
---- -------- -------- -------- -------- -------- ------- -------
-------
50 3001134 3001557 2820471 3161853 3167160 0.99986 1.06405
0.94917
100 5761455 5762209 5428681 6088469 6095968 0.99987 1.06130
0.94629
150 8444396 8445138 7967642 8941792 8947016 0.99991 1.05984
0.94437
200 11078937 11079974 10463628 11742238 11749807 0.99991 1.05880
0.94351
250 13679318 13680169 12928601 14500586 14517771 0.99994 1.05807
0.94336
300 16252325 16253409 15369409 17248950 17258600 0.99993 1.05745
0.94222
350 18803526 18804657 17790479 19962114 19977267 0.99994 1.05694
0.94196
400 21336326 21337378 20194905 22664825 22677242 0.99995 1.05652
0.94138
450 23853038 23854118 22584966 25346170 25361087 0.99995 1.05615
0.94109
500 26355867 26356832 24962408 28011558 28030761 0.99996 1.05582
0.94089
550 28845356 28847058 27328610 30667682 30687814 0.99994 1.05550
0.94058
600 31324703 31326045 29684688 33306524 33333500 0.99996 1.05525
0.94050
650 33793395 33794819 32031565 35948151 35968852 0.99996 1.05500
0.94006
700 36252931 36254242 34370013 38570624 38594739 0.99996 1.05478
0.93991
750 38703181 38705046 36700688 41192229 41211899 0.99995 1.05456
0.93957
800 41146179 41147862 39024157 43801214 43820966 0.99996 1.05438
0.93938
850 43581966 43583236 41340910 46394745 46422492 0.99997 1.05421
0.93937
900 46009215 46011648 43651379 48997675 49016961 0.99995 1.05402
0.93901
950 48431471 48433522 45955943 51588931 51604800 0.99996 1.05387
0.93880
1000 50847534 50849234 48254942 54166822 54186390 0.99997 1.05373
0.93872
Speaks for itself, right? pi/(x/ln(x)) is moving slowly toward 1,
pi/jsh
drifts slowly farther from 1, the Merten's approximation to `jsh` is
good,
and li(x) is by far the best estimate across this range. Nothing here
is
surprising (well, except possibly to you, since the primes demonstrably
don't act the way you've talked yourself into believing they act).
This isn't going to get better as x increases, because the PNT has been
/proved/: there's no serious doubt about it. Check the limit of the
Wikipedia table, at x=10^23. There:
pi/li ~= 0.999999999996
pi/(x/ln(x)) ~= 1.01964
pi/mertens ~= 0.908026
and
mertens/(x/ln(x)) ~= 1.123
So everything is still going the way century-old theory predicts at
x=10^23.
If you don't believe the Mertens approximation there, you can multiply
1-1/p
across the 12431880460 primes <= sqrt(10^23) yourself ;-)
-----------------------------------------------------------------------------------------------------------
Felix of The North
> > So I did some more research and noticed that I'm not the first person
> > to notice that
> >
> > ((p_j - 1)/p)*...*(1/2)
James - if you would like to make a contribution to what will be called
"Harris Space", you are invited to leave feedback on this blog. You will be
named as a coauthor for your contributions, and it might even get published.
http://sciphysicsopenmanuscript.blogspot.com/
The integration method outlined in this blog may become known as the Harris
Integral.
Actually, I really do need help with this. I need to search for prior work
prior to submission and I do not have access to a decent library at this
time.
jst...@msn.com
Cool! I wanted a table!
Well, it clearly looks like I was wrong on several positions, but the
truth is what I was looking for here, and now there are more puzzles to
consider.
It turns out that you can rigorously consider the approach I outlined
using a method attributed to Legendre's and prove beyond any doubt that
the approach is just not using something called the floor() function,
and not correcting for the count of primes up to and including sqrt(x)
or the number 1, as it's not considered prime.
If you do that with this approach it is perfectly exact, though that is
not practical, but it does offer answers.
It says that the difference between the prime distribution and various
functions is mostly about remainders!!!
For readers who don't know what floor() is, floor(3.1415) = 3, as it
just means throww of any remainder and go with the integer.
For example, up to 10, you have
10 - floor(10/2) - floor(10/3) + floor(10/6) + 2 - 1 = 4 primes
which is explained as, subtract the evens from 10, and subtract the
numbers divisible by 3, but then you are also subtracting some evens
that have 3 as a factor so add back in those that have both as a
factor, add 2 for the primes 2 and 3 being subtracted and subtract 1,
since 1 is not prime. Divide 4 by 10 and you have 25% probability of
primeness.
Well, looking at 2/3 as the odds a number does NOT have 3 as a factor
and 1/2 as the odds it does not have 2 as a factor, I have the
equivalent of
10 - 10/2 - 10/3 + 10/6 = 10(1 - 1/2 - 1/3 + 1/6) = 10*(1/3)
which shows that this idea is really just like going through the same
process as above, but without using floor() so there are remainders
floating around, and without correcting for 2 and 3 being prime, as
well as 1 not being prime.
So you can figure it all out without probability.
And the mathematics is trivial and absolute, while perfectly explaining
everything there is to know about how the prime distribution behaves.
Every feature can be explained using this information, so what gives
with mathematicians not explaining simply?
Thanks again for the table.
James Harris
Jim Lanson
<jst...@msn.com> wrote in message
news:1155862461....@p79g2000cwp.googlegroups.com...
> jshs...@yahoo.com wrote:
>> jst...@msn.com wrote:
>
> For readers who don't know what floor() is, floor(3.1415) = 3, as it
> just means throww of any remainder and go with the integer.
>
Nice, "JSH discovers truncation".
Perhaps part of JSH was truncated when he was a child.
jst...@msn.com
And that's the problem with math society. No problem if you think I'm
being refuted, but if I come back with the counter, and the real answer
on primes, you go for the low-blow.
Hey, it's not my fault. Primes are just this way. Human needs do not
trump mathematical truth.
It's the mathematical reality.
I know many of you hate simplicity, and now may hate prime numbers for
their simplicity, but the mathematics is just the way it is, the way it
always has been, the way it always will be.
You can just use the Legendre's ideas, drop use of floor() and come up
with the equations that follow from the probabilistic angle, proving
that the difference between the exact count of primes and these other
functions is mainly about the remainder.
My point is that mathematicians don't want THE answer.
They want complexity, and something hard to explain that is convoluted
with the opportunity for lots of books and papers or I wouldn't be the
one explaining primes now.
It'd have been done a hundred years ago, but wait, didn't the idea of
"pure math" where useless, non-practical results are cheered come up
about a hundred years ago?
James Harris
jshs...@yahoo.com
jst...@msn.com wrote:
> Thanks again for the table.
>
> James Harris
No problem. It provided you with yet another opportunity to make a fool
of yourself.
jst...@msn.com
Nope. I connected the dots. I needed a table to be sure, but didn't
want to calculate one myself. So you and Peters provided one for me.
Thanks.
I'm a researcher who comes up with a lot of ideas. Testing them all
myself would take too much time, so I get other people to test many of
them for me.
That leaves me time to concentrate on the full explanation which I now
have.
I now completely understand the prime distribution, and Li(x) and x/(ln
x) just come into the picture because of this neat thing with the
floor() function. The "error" is just a remainder error.
You are in a world where I need information, and people like you can
bring it to my attention, so as I need, you do.
Secondary impacts are unimportant, and to be expected as you
rationalize explanations where you are in control of your own behavior,
when it looks to me like, I am.
Good job. You may be needed in the future. I have no doubt that you
will hang around, as have others before you, until they burnt out.
James Harris
jst...@msn.com
Obviously an attempt at reverse psychology, but hey! It might work!
I figured it was so obvious I might as well note that in reply.
But still, hey, it might work!
Reality is that yeah, I was way off with what I said in my original
post, but the saving grace from my perspective is figuring out how
everything does connect, and yes, it is all about the floor() function.
That answer is clear. The prime distribution is not complicated in
terms of the 'why' of its behavior. Not complicated at all.
James Harris
Jim Lanson
<jst...@msn.com> wrote in message
news:1155870739....@m79g2000cwm.googlegroups.com...
> Jim Lanson wrote:
>> <jst...@msn.com> wrote in message
>> news:1155862461....@p79g2000cwp.googlegroups.com...
>> > jshs...@yahoo.com wrote:
>> >> jst...@msn.com wrote:
>> >
>> > For readers who don't know what floor() is, floor(3.1415) = 3, as it
>> > just means throww of any remainder and go with the integer.
>> >
>>
>>
>> Nice, "JSH discovers truncation".
>> Perhaps part of JSH was truncated when he was a child.
>
> And that's the problem with math society. No problem if you think I'm
> being refuted, but if I come back with the counter, and the real answer
> on primes, you go for the low-blow.
you just discovered the floor function, and it has no real part in what you
are attempting. But you do not see that.
>
> You can just use the Legendre's ideas, drop use of floor() and come up
> with the equations that follow from the probabilistic angle, proving
> that the difference between the exact count of primes and these other
> functions is mainly about the remainder.
>
> My point is that mathematicians don't want THE answer.
WRONG - the answer in many cases it too complex to discribe, and many have
fuzzy answers, wait till you get to the hard problems.
>
> They want complexity, and something hard to explain that is convoluted
> with the opportunity for lots of books and papers or I wouldn't be the
> one explaining primes now.
Most all of us learned about this long ago, although it is new discovery to
you!
and there is a lot of good work out there, you should read it.
jshs...@yahoo.com
Hmmmm. I seem to remember how you said awhile back that you wouldn't be
responding to my posts anymore because I am just a troll and a
parasite, yet here you are these past few days responding to my posts.
I wonder who is controlling who.
Of course I will provide with something that will allow you to keep me
entertained. You have been doing that for a couple years now. How one
person can make such a fool out of himself time and time again, with
such utter bravado is a nice amusement.
Tim Peters
> Nope. I connected the dots. I needed a table to be sure, but didn't
> want to calculate one myself. So you and Peters provided one for me.
> Thanks.
>
> I'm a researcher who comes up with a lot of ideas. Testing them all
> myself would take too much time, so I get other people to test many of
> them for me.
>
> That leaves me time to concentrate on the full explanation which I now
> have.
>
> I now completely understand the prime distribution,
LOL! You're actually a bit behind Eratosthenes at this point, so have about
2000 years to catch up on :-)
> and Li(x) and x/(ln x) just come into the picture because of this neat
> thing with the floor() function. The "error" is just a remainder error.
Cool! Then you should be able to quantify it, and give the /true/ error
term for the PNT (which would be worth a million dollars to you if you can
convince the Clay Mathematics Institute you've solved it).
> You are in a world where I need information, and people like you can
> bring it to my attention, so as I need, you do.
>
> Secondary impacts are unimportant, and to be expected as you
> rationalize explanations where you are in control of your own behavior,
> when it looks to me like, I am.
>
> Good job. You may be needed in the future. I have no doubt that you
> will hang around, as have others before you, until they burnt out.
You're on an exceptional roll this week.
Proginoskes
jst...@msn.com wrote:
> So I did some more research and noticed that I'm not the first person
> to notice that
No, people did the research for you. Stop hogging credit.
--- Christopher Heckman
> [The Shovel has removed the rest of this post.]
Proginoskes
jst...@msn.com wrote:
> [...]
> Well, it clearly looks like I was wrong on several positions,
All right ... Who are you, and what have you done to the real James
Harris???
> but the
> truth is what I was looking for here, and now there are more puzzles to
> consider.
>
> It turns out that you can rigorously consider the approach I outlined
> using a method attributed to Legendre's and prove beyond any doubt that
> the approach is just not using something called the floor() function,
> and not correcting for the count of primes up to and including sqrt(x)
> or the number 1, as it's not considered prime.
>
> If you do that with this approach it is perfectly exact, though that is
> not practical, but it does offer answers.
>
> It says that the difference between the prime distribution and various
> functions is mostly about remainders!!!
>
> For readers who don't know what floor() is, floor(3.1415) = 3, as it
> just means throww of any remainder and go with the integer.
>
> For example, up to 10, you have
>
> 10 - floor(10/2) - floor(10/3) + floor(10/6) + 2 - 1 = 4 primes
Congratulations. You've finally come around to the inclusion-exclusion
formula that I posted over a year ago, and which you called worthless.
> which is explained as, subtract the evens from 10, and subtract the
> numbers divisible by 3, but then you are also subtracting some evens
> that have 3 as a factor so add back in those that have both as a
> factor, add 2 for the primes 2 and 3 being subtracted and subtract 1,
> since 1 is not prime. Divide 4 by 10 and you have 25% probability of
> primeness.
>
> Well, looking at 2/3 as the odds a number does NOT have 3 as a factor
> and 1/2 as the odds it does not have 2 as a factor, I have the
> equivalent of
>
> 10 - 10/2 - 10/3 + 10/6 = 10(1 - 1/2 - 1/3 + 1/6) = 10*(1/3)
>
> which shows that this idea is really just like going through the same
> process as above, but without using floor() so there are remainders
> floating around, and without correcting for 2 and 3 being prime, as
> well as 1 not being prime.
>
> So you can figure it all out without probability.
But the downside is that you need a list of the actual primes, this
time <= sqrt(N).
> And the mathematics is trivial and absolute, while perfectly explaining
> everything there is to know about how the prime distribution behaves.
Not quite. As I've mentioned before, there are many prime-counting
*formulas* (I can recall at least 4 right off), but none of them have
settled the question why the primes are distributed the way they are.
None of them can answer the Riemann Hypothesis one way or another, for
instance.
--- Christopher Heckman
Moshira
<jst...@msn.com> wrote in message
news:1155873277.5...@74g2000cwt.googlegroups.com...
> jshs...@yahoo.com wrote:
>> jst...@msn.com wrote:
Checkout this crank/crackpot
http://www.crank.net/harris.html
and his blog
http://jstevh.blogspot.com/2006/06/jsh-i-will-get-my-money.html
jst...@msn.com
> <jst...@msn.com> wrote in message
> news:1155873277.5...@74g2000cwt.googlegroups.com...
> > jshs...@yahoo.com wrote:
> >> jst...@msn.com wrote:
>
> Checkout this crank/crackpot
>
That's not my blog.
James Harris
W. Dale Hall
However, it is a word-for-word quotation of an article you wrote, on
sci.math, and which appeared on Nov 23, 2003.
Try to deny that.
Dale