Re: JSH: Math journals do not just die

....

[jstevh@msn]
Last I checked you were making statistical arguments.

In part, yes; in part, no.

However, with prime numbers an excellent example of an area where
things are not completely random is the prime distribution itself,

More accurate to say they're "completely non-random".

where 1/(ln x) rules, and that is an example of a guiding equation.

1/ln(x) doesn't "rule" anything. A non-trivial theorem /relates/ the
distribution of primes to the curve 1/ln(x), which is a derived result,
not a guiding principle (or an /a priori/ "rule").

Otherwise there is no REASON for the behavior mathematically.

Gibberish.

Part of the problem here is that mathematicians have routinely gotten
away with mathematical gibberish as an argument style, where by
babbling enough of what I call math-ese you can b.s. your way through
and convince people.

True or false, that's none of the problem /here/. The problem here is
that you don't know anything about the topic at hand, and steadfastly
refuse to learn.

So I'm bottom-lining it for people who know that in mathematics you
have equations.

LOL. Yes, you do.

If you are right, give the equations.

"The equations" for /what/, specifically? If you want to know "why"
the residue pairs

<1, 1>
<1, 2>
<2, 1>
<2, 2>

distribute as they do up to a given x, I've explained that several
times already. It's entirely about the distribution of prime gaps, and
you can use Brent's formulas in:

"The Distribution of Small Gaps Between Successive Primes"
http://wwwmaths.anu.edu.au/~brent/pd/rpb021.pdf

to get good predictions of how those pairs distribute up to a given x.
Computationally they work very well, although they build on conjectures
due to Hardy & Littlewood from the 1920's that still haven't been
proved. It's a consequence of the formulas that the pairs <1, 1> and
<2, 2> appear through "small" x far less often than they would if the
sequence were truly random. That wholly matches what everyone who got
off their ass (leaves you out) long enough to check observed.

It's also a consequence that "small" here extends at least through
primes with a few hundred bits. Also that the sequence can never act
truly randomly through any finite x, no matter how large. I would bet
that asymptotic ("at infinity") equidistribution also follows from the
formulas, but haven't been able to show that rigorously.

None of this is new: I've explained it several times already, and as I
said last time, you ignore substantive replies. You still have no
explanation whatsoever for the behavior people observed, while Brent's
formulas can be used to make testable quantitative predictions. You're
at least a week behind here, still spinning your wheels at the starting
line. Catch up or give it up.

...
I emphasize this as it's crucial to make the point that something is
wrong here,

Have a mirror?

so that people do NOT simply decide that if you keep
disagreeing with me, I must be wrong, and you right, which is an
unfortunate default that often takes place.

I want them to notice you are not really answering the question.

As above, you don't ask specific questions, just deliever rants and
make unfounded accusations. How many times do you need a reference to
Brent's paper before its supreme relevance sinks in? Try reading it
(no, not skimming it with a brain clouded by anger, desperate to "get
even", but /reading/ it dispassionately, as the piece of mathematics it
is).

...
I say, if you are not just trying to convince people of your position,

No need for that -- the argument based on Brent's formulas is clear.
People are free to check the math themselves, several have, and there's
no disagreement about it among those who have.

Now I suspect you /did/ look at Brent's paper, but weren't able to
understand it. That would be fine if you said you needed help with it,
but you'll never do that. Instead you'll continue bluffing. /That/
earns contempt, not that you simply don't understand the references
given to you -- you don't care spit about learning or truth.

then you can give some freaking equations

Already done, multiple times.

or something versus denial of randomness

Still relevant: your sequence remains non-random, and /obviously/ so,
by any accepted meaning of the word. No "freaking equations" are
needed to demonstrate that beyond the one you started with: mod(p, 3).

or going to statistical arguments.

Beyond moronic. Talking about prediction and correlation /necessarily/
invokes statistical arguments -- they're statistical concepts.

I'll give yet another example:

Say you flip a coin 10 times and it comes up head 10 times.

By statistical tests the coin is probably flawed, right?

Definitely not, and this has been explained multiple times too. I'm
not going to repeat it again.

But how can you be sure?

Well, check the damn coin!

Statistics can point in a direction, but it cannot prove.

When probabilities are smaller than 1 in 10^60000, as they /have/ been
in some computational experiments here, a sane person draws the obvious
conclusion.

Is that absolute proof? Of course not. The absolute proof was that
your sequence is defined by a computable formula, and nothing beyond
that was truly needed. That's "checking the coin" here, and it was
absolutely known to be non-random before trying /any/ computational
tests. The latter were just interesting to perform, to see how the
non-randomness might manifest via one of the simplest standard
statistical tests for randomness. That brought up some interesting
math, in fact. Unfortunately, you don't know what it was; fortunately,
if you care to find out, I don't delete my posts from archives, so it's
still there waiting for you to notice.

You actually have to go in and prove something one way or the other.

Already done.

If p mod 3 is not random then there is a RULE or there are RULES which
would be embodied in MATHEMATICIAL EQUATIONS that would show that it's
not.

Already done.

As an example, the prime distribution--the count of primes--shows
non-randomness in its close relation to 1/(ln x).

If the Hardy-Littlewood conjectures are true, the distribution of prime
gaps follows very similar formulas. See Brent for details. In general
the conjectures give that the number of primes <= x with gap g
asymptotically equals the sum of

x * C_g_k * integral of dt/ln(t)^(k+1) from 2 to x

where k ranges from 1 to g/2, and C_g_k are constants depending on g
and k. That in turn is asymptotically equal to the sum of

x * C_g_k/ln(x)^(k+1)

That explains the distribution of residue pairs actually seen, and
/quantitatively/ explains how <1, 1> and <2, 2> are doomed to appear
"far less often than their fair share" through primes with at least a
few hundred bits.

But none of that was necessary to show non-randomness. Non-randomness
was immediate.

.... [more old stuff; don't care about it myself] ...

.

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