from Jeffrey Cook, the author of the Riemann Hypothesis paper I

discussed earlier.

-------- Original Message --------

Subject: Re: At last, a proof that RH is true

Date: Sat, 10 Jan 2009 20:05:07 -0000

From: Jeffrey N Cook <antidyne@...>

To: Alan Eliasen <eliasen@...>

Hi, Alan. This is Jeff Cook, author of the paper and purported proof

in question. What you have found in my paper regarding u (x) is

valid, as it is misrepresented. On 5-21-08 I noticed the values of

this function as written were not matching my recorded worksheet

values. I had it noted at http://www.jeffreyncook.com/jeff%20cook%

20updates.htm that there was an error in the paper. I noted that

equation (83) had a misprint. However, in a hurry, I looked over the

equation alongside my worksheet and noticed that a few decimal points

on lower values of x were not matching what I originally had in the

paper. I could not find the problem immediately and ignored it for

the time being and simply changed the values in the paper that did

NOT match my worksheet. This was a foolish mistake. On 5-28-08 I

had it noted there was no mistake afterall. But the mistake is still

there.

Alan Eliasen`s: brief points on a couple major places where [Cook`s

purported proof is] incorrect."

Note the x^x in eq. 84 above, which grows rapidly. From

these meager data points, the paper claims (eq. 88) that the maximum

error J[x]-Prime[x] *for all x* is on the order of 12. This is

completely wrong."

[W]hen you get to even x=206, the error is -19.6006."

[T]hen the error just starts to increase linearly (actually,

a little worse than linearly.)"

Thus, the completely unsupported assertions about the error

of this prime-approximating function in eq. 87-89 are wrong, as they

seem to be based on incorrect ideas of how the functions actually

behave."

[T]he leaps of faith from one unsupported equation to the

next in this chain are immense.) There is of course no reason to

believe the extrapolations to infinity, as the assertions are wrong

for even very small numbers."

u(x) = log_e (Pi * i * y * x^2 * Beta (x) * A (x))

u(y) ~ log_e (-Pi * x^2 * Beta (x) * A (x))

Lim y = 1 / pi to infinity

u(x) in the paper is derived from u(y). The only reason why I keep

the pi in there is to clear the first few values. Eventually, the

value of Pi * i * y = -1 in accordance with the rest of the paper.

However, u(x) is not correct (missing a variable) that is not

included.

I thank you for going over my results in order to bring me back to

this. The problem is that there is a missing variable in (83) that I

lost in my notes. The values in my work are accurate (but not

reflected in my paper). I will dig out my old equations and fix this

in the next day or so. It is only a typo, a missing variable.

In any case, this equation is part of the second proof. While ugly

indeed, and needing to be fixed, the first proof still stands. You

wrote:

In addition, the definitions of things like the log integral

(eq. 40) don't seem to be the definition of the log integral that I'm

familiar with. It appears to be a summation of discrete terms, and

not the integral! I may just not recognize it in this form, but it

doesn't seem right at all. If that's important in the paper, it's

another problem."

This is not a problem and is commonly understood by those familiar

with the Prime Number Theorem. Pi (N) ~ N / log (N) is the older and

obsolete Prime Number Theorem. Now we Express it as pi (x) ~ Li

(x). The derivative of log (x) is 1 / x, so log (x) is the

integral. The inverse of a derivative is an integral and the inverse

of differentiation is integration. So long as N does not equal -1,

the integral x^N is x^(N+1)/(N+1). But there is no common function

that can be used to express the integral of 1 / log (x), Li (x) takes

on the area under the graph of 1 / log (x) or the integral of the

inverse of log x with respect to x, from zero to infinity. Integrals

and sums are very closely linked. [7]

[7] from my paper is Derbyshire`s book.

Remember, there are two proofs of the RH in the paper. The error

involving the second will be fixed soon.

Thanks,

Jeff

--- In primenumbers@yahoogroups.com, Alan Eliasen <eliasen@...> wrote:

>

> Alex Petty wrote:

> > My colleague has settled the long outstanding question of

Reimann's [sic]

> > Hypothesis and shown conclusively that all non trivial zeros of

the

> > zeta function do indeed have Real part one half, ie. the

hypothesis

> > has been proven to be true. To review this proof, now in pre-

print,

> > please follow the url:

> >

> >

http://www.singularics.com/science/mathematics/OnNeutronicFunctions.pd

f

>

> I don't want to spend a lot of time on this, so I'll be brief and

> point out a couple major places where it's incorrect. The paper

claims

> some amazing things, for example, "These functions are put together

to

> reveal a new function whose difference from the Prime Number

Function

> (2, 3, 5, 7, 11...) to infinity is zero..."

>

> This amazing function is very simple and can be summarized as the

> following:

>

> beta[x] := 6x + 1 (rearrangement of eq. 74)

>

> r[x] := 4x/5 - 15 (eq. 82)

>

> A[x] := 1 - 6x^x (eq. 84)

>

> u[x] := ln[-pi beta[x] A[x]] (eq. 83)

>

> J[x] := r[x] + u[x] (eq. 86)

>

> Where J[x] is supposed to be an approximation of the function

that

> lists the primes, (I'll call it Prime[x]) e.g. 2,3,5,7,11. That is,

> Prime[1]=2 Prime[2]=3, etc.

>

> From this, I could reproduce the values in the table 27. No

> discrepancy there. The paper goes on to graph the first 140 terms

of

> this to show how well J[x] matches Prime[x]. (I'll make a note,

though,

> that when you're talking about the primes, looking only at the

first 140

> terms of a sequence and extrapolating beyond that is a sure recipe

for

> disaster.)

>

> But why only the first 140 terms? Probably because 64-bit IEEE-

754

> floating-point hardware overflows after this! Note the x^x in eq.

84

> above, which grows rapidly. From these meager data points, the

paper

> claims (eq. 88) that the maximum error J[x]-Prime[x] *for all x* is

on

> the order of 12. This is completely wrong.

>

> If you use a real computing environment to evaluate larger

numbers,

> the error does bounce around and stay smaller than 12 for very small

> values of x, but when you get to even x=206, the error is -

19.6006. And

> then the error just starts to increase linearly (actually, a little

> worse than linearly.) At x=1400, the error is -398.109. At

x=10000,

> the error is -4626.66. At x=20000, the error is -10667.6. And the

> trend continues. That's a lot larger than 12, but the paper says

that

> "the maximum value of all error terms from 1 to infinity becomes

very

> clearly" 12. That is no longer clear.

>

> Thus, the completely unsupported assertions about the error of

this

> prime-approximating function in eq. 87-89 are wrong, as they seem

to be

> based on incorrect ideas of how the functions actually behave.

(And the

> leaps of faith from one unsupported equation to the next in this

chain

> are immense.) There is of course no reason to believe the

extrapolations

> to infinity, as the assertions are wrong for even very small

numbers.

>

> I don't know if anything else in the paper relies on this, but it

> shows that the numbers weren't checked even to moderately-sized

values,

> and assertions about the primes are incorrect.

>

> In addition, the definitions of things like the log integral

(eq. 40)

> don't seem to be the definition of the log integral that I'm

familiar

> with. It appears to be a summation of discrete terms, and not the

> integral! I may just not recognize it in this form, but it doesn't

seem

> right at all. If that's important in the paper, it's another

problem.

>

> My projection of the veracity of claims about the Riemann

Hypothesis

> are thus approximately epsilon. But this is probably sufficient for

> this list.

>

> --

> Alan Eliasen | "Furious activity is no substitute

> eliasen@... | for understanding."

> http://futureboy.us/ | --H.H. Williams

>

--

Alan Eliasen | "Furious activity is no substitute

eliasen@... | for understanding."

http://futureboy.us/ | --H.H. Williams