Re: Pi(x) - How many primes below x?

[Anonymous <nobody@replay.com>]

Here are a couple of overdue Choate blooper corrections.

Regarding the Prime Number Theorem, Choate originally wrote:

> I had typed x/ln(x) as the asymptotic limit for the number of primes less
> than x.
> 
> This is incorrect. It should be,
> 
> x/log(x)

It should be noted that ln(x) is the logarithm to the base e of x.
log(x) is somewhat ambiguous as to the base, but when it is contrasted
with ln(x) as Choate does here, it implies that the base is 10.  In
fact the correct formula uses the base e, and x/ln(x) is not "incorrect"
as Choate is claiming.

When Choate's error was pointed out, he responded by quoting
http://www.utm.edu/research/primes/howmany.shtml, which says that the
formula is x/log x.  What Choate failed to notice is that the web page
clearly states that its logs are to the base e.  In other words, the
"log x" on that page is equivalent to the "ln(x)" which Choate originally
wrote.  Choate's original formula was the right one, and in writing that
his formula was incorrect, he only displays his own confusion.


With regard to his ludicrous model of a spark gap inside a conductive
sphere, Choate originally wrote:

> The spark gap generates sparks and that builds up free electrons in the
> space inside the sphere (whether it is gas filled or a vacuum is
> irrelevant). As that charge builds up it will be all of one type, electrons.
> Now the electrons repel each other and therefor move in a circular motion
> with the spark gap as the center. They strike the surface of the sphere and
> tunnel through to the outside surface where they reside. The amount of
> charge at any one point is related to the curvature of the surface at that
> point. Since a sphere is constant curvature the charge will be evenly
> distributed. It will continue to build up so long as you supply power to the
> spark gap. In an ideal world it will get bigger and bigger. In the real
> world at some point insulation breaks down and normal current flow takes
> place.

He believes that the spark gap emits electrons, which strike the inside
surface of the sphere and "tunnel through" to the outside.  He says that
the charge will continue to build up so long as you supply power to the
spark gap.

Here is another message in which he made the same point:

> This is in addition to the charge that steadily builds up in the shell as
> the electrons accrete over time. This can be modelled with an integral of
> the flow rate of the current in the battery (it after all is Coulombs/s).
> It's not too hard (k * I). (I'm not going to go into what happens as the
> charge on the shell builds up as we're discussing here the applicability of
> wave equations as a reliable model).
> 
> So what do you get? A hell of a charge that will go bang at some point when
> some insulation give way.

When it was pointed out how ignorant this idea was, and how it violates
Gauss's Law, Choate tried to backpedal by proposing that the battery
itself was charged in the first place.  He even drew a picture:

>                2N e-          1N p+
>
>                     |       |
>                     |       |
>       |-------------|       |--------------|
>       |             |       |              |
>       |             |       |              |
>       |                                    |
>       |                                    |
>       |------0 0------------------0     0--|
>                                    \
>                                      \
>           spark gap               switch

It should be obvious that this device fails to generate the phenomena which
he describes above.  In the first place, since it has a net negative
charge, the charge will appear on the outside of the sphere BEFORE THE
SWITCH IS THROWN.  There is no spark active, no electrons being emitted,
yet a negative charge appears on the outside of the conductive sphere.
This is an elementary application of Gauss's Law.

Then, when the switch is thrown, there will be a spark, and some of
the charges will neutralize each other, but of course the net charge
will stay the same, by conservation of charge.  The phenomenon Choate
described of electrons being emitted by the spark gap, moving outward
and striking the sphere, and then tunnelling through to make charge
appear on the outside, will occur only in Choate's deluded imagination.

Furthermore, there will be no "build up" of charge.  The charge will be
there from the moment the device is put into the sphere, long before
the switch is thrown.  Throwing the switch will have NO EFFECT WHATSOEVER
on the outside of the sphere.

Michael Motyka spent several days trying to patiently explain all this
to Choate, to no avail.  He finally gave up in frustration.  Choate is
almost completely immune to enlightenment.  He has certainly proved that
patience and politeness make no dent in his thick skull.  We shall see
whether blunt frankness is any more effective.