Re: Primes, probability and politics

[jstevh@xxxxxxx]
...
So the probability that x does NOT have p as a factor is 1 - 1/p =
(p-1)/p, and for each prime you include you just need to multiply the
probabilities.

So, trivially, you have the probability that x is prime:

probPrime(x) = ((p_j -1)/p_j)*...*(1/2)

where there are j primes up to sqrt(x) and p_j is the jth prime.

[jankrihau@xxxxxxxxxxx]
Trivial to you, but wrong. The left hand side is asymptotically
1/ln(x),
the right side is asymptotically 2 exp(-gamma) / ln(x),

Where did "2" come from? I don't believe that part.

where gamma is the Euler-Mascheroni constant 0.57721566... This
follows from Merten's theorem, and yes, I do have a reference:

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

[jstevh@xxxxxxx]
More detail please... I followed the link and am suspicious of your
assertion.

Heh. You also ignored me last week when I pointed out the footnote in Hardy
& Wright warning:

Considerations of this kind explain why the usual "probability"
arguments lead to the wrong asymptotic value for pi(x).

and urged you to complete the exercise of figuring out what asymptotic value
your argument /does/ lead to. In fact, you poo-poo'ed me twice on that, so
I gave up trying. More, *** Winter urged you to look up Mertens's Theorem
even earlier, the very first time you tried making this argument. You
ignored him too, IIRC. You can't say you weren't warned.

Worse, I think that what I call math-ese is a peculiarly powerful
weapon in the arsenal of math people who are, well, not telling the
truth.

Do you need to act like a total ass every time?

You put up some complex assertion, which you figure is hard to
parse through it, and hope that no one bothers to really check you on
it.

No, what he said was easy to follow for someone who can read math. I'm not
sure on where you're getting confused, because you made no effort to explain
where you stopped following the argument. So I can only assume you have no
idea whatsoever about what anything he said meant.

I'll try. An equivalent form of Mertens's theorem appears as Theorem 429 in
Hardy & Wright:

the product over all prime p <= x of 1 - 1/p
is aymptotically equal to
e^-gamma / ln(x)

To get the form on MathWorld, take the reciprocal of both sides, then divide
both sides by ln(x), then replace x by p_n. Same thing, but I won't say
more about that because the Hardy & Wright version is applicable as-is to
your argument, just by noting that

(p_j -1)/p_j = 1 - 1/p_j

via simple algebra. Your probPrime(x) expression is exactly the left-hand
side of the H&W form of Mertens's theorem.

BTW, if you don't believe me either, you can also find the H&W spelling on
Chris Caldwell's prime pages:

http://primes.utm.edu/glossary/page.php?sort=MertensTheorem

And people not able or willing to check assume you are right, so you
can put up a false assertion and be believed based on the intellectual
weakness of others.

How about yours? You have a computer, and you could have /checked/ your
argument against reality very easily here. Let's take a look at each
10000th prime. Here, where

normal: the usual approximation x/(ln(x) - 1) to pi(x)
x*JSH: x * product across all prime p <= x of (p-1)/p
Mertens: x * e^-gamma / ln(x)

x pi(x) normal JSH Mertens
-------- ------ ------ ----- -------
104729 10000 9918 5085 5086
224737 20000 19848 10237 10239
350377 30000 29776 15406 15408
479909 40000 39723 20595 20597
611953 50000 49653 25783 25786
746773 60000 59629 31002 31003
882377 70000 69531 36183 36187
1020379 80000 79495 41404 41407
1159523 90000 89445 46620 46623
1299709 100000 99383 51833 51836
1441049 110000 109328 57051 57055
1583539 120000 119285 62279 62282
1726943 130000 129244 67510 67512
1870667 140000 139167 72722 72726
2015177 150000 149093 77938 77943
2160553 160000 159029 83161 83166
2307229 170000 169008 88410 88414
2454587 180000 178990 93662 93665
2601857 190000 188927 98890 98894
2750159 200000 198895 104136 104140
2898527 210000 208831 109366 109370
3047767 220000 218792 114611 114615
3196933 230000 228716 119836 119841
3346601 240000 238643 125064 125069
3497861 250000 248645 130335 130339
3648923 260000 258606 135584 135587
3800201 270000 268554 140826 140830
3951161 280000 278456 146043 146049
4103629 290000 288431 151302 151307
4256233 300000 298392 156554 156559
4410317 310000 308425 161847 161849
4562693 320000 318325 167066 167070
4716053 330000 328267 172310 172314
4869863 340000 338218 177559 177563
5023307 350000 348125 182784 182790
5178049 360000 358096 188045 188051
5332519 370000 368030 193287 193293
5487701 380000 377992 198544 198551
5644031 390000 388009 203834 203838
5800079 400000 397991 209104 209107
5955031 410000 407885 214325 214330
6111613 420000 417868 219596 219601
6268289 430000 427839 224862 224866
6424937 440000 437794 230119 230123
6581963 450000 447757 235380 235384
6738889 460000 457699 240631 240635
6895393 470000 467599 245858 245865
7052113 480000 477499 251086 251094
7210759 490000 487507 256375 256381
7368787 500000 497461 261634 261641
7528121 510000 507485 266933 266937
7685801 520000 517392 272166 272172
7844731 530000 527364 277436 277442
8003537 540000 537316 282696 282702
8163047 550000 547299 287973 287978
8322241 560000 557251 293233 293239
8482259 570000 567242 298516 298520
8640679 580000 577122 303736 303743
8799919 590000 587042 308979 308988
8960453 600000 597031 314262 314269
9121439 610000 607038 319554 319560
9281011 620000 616945 324791 324799
9442229 630000 626944 330080 330086
9602443 640000 636871 335329 335336
9763393 650000 646832 340598 340605
9925439 660000 656852 345899 345904
10086767 670000 666817 351170 351175

Do it yourself if you don't believe me: Mertens's theorem clearly gives an
excellent approximation to your guess, while both of those suck as
approximations to pi(x).

So, step out your claim, please.

See above.

If I am wrong about your claim and you can validate it, I apologize up
front for coming out and saying you could be lying and deliberately
trying to fool people in a way that has worked for mathematicians in
the past.

Given that you appeared to make no effort at all to check one way or the
other, I suggest that bringing up the possibility he may be lying was
inexcusable ... yup, I officially retire from trying to give you a serious
answer about anything anymore. Congratulations :-)


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