Re: [PrimeNumbers] At last, a proof that RH is true
- Alex Petty wrote:
> My colleague has settled the long outstanding question of Reimann's [sic]I don't want to spend a lot of time on this, so I'll be brief and
> 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:
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
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=2 Prime=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
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
Alan Eliasen | "Furious activity is no substitute
eliasen@... | for understanding."
http://futureboy.us/ | --H.H. Williams