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From: OsherD on 27 Sep 2007 18:42
>From Osher Doctorow

Andrea Ossicini of Roma Italy in arXiv: 0709.4173 v2 [math-GM] 27 Sep
2007, 5 pages, introduces a special function AO with some remarkable
properties:

1) AO(1 -s) = gamma)(1 - s)zeta(1 - s)L(1 - s)/k^(1 - s)

where k is the number pi (not related to PI as Probable Causation/
Influence) and L is Dirichlet's L function:

2) L(1 - s) = (2/pi)^s gamma(s) sin(ks/2)L(s)
3) L(s) = sum (-1)^n/(2n + 1)^s, sum over n = 0 to infinity

It turns out that:

4) AO(s) = A)(1 - s)

a theorem proven by Ossicini, so the complex zeros of AO(s) are all
localized in the strip determined by Re(s) in (0, 1} so the nontrivial
zeros of zeta(s) function are perfectly equal to those of the AO(s)
function.

The Probable Causation/Influence (PI) has similar properties at its
extremes s = 0 and s = 1, or more precisely at the two-variable
argument generalizing the one-variable s. That is to say:

5) P(X-->Y)(1, 1) = P(X-->Y)(1 - 1, 1 - 1) = P(X-->Y)(0, 0)

which can be seen from the fact that:

6) P(X-->Y)(x, y) = 1 + F(x, y) - FX(x)

where F(x, y) is the joint cumulative distribution function (cdf) of X
and Y and FX(x) in the univariate (marginal) cdf of X for X, Y
continuous random variables. The result (5) follows from the fact
that, normalizing the ranges of X and Y to [0, 1] or [0, 1] x [0, 1],
or letting 1 represent infinity if X for example is unbounded above,
and similarly for lower bound 0, we have F(1, 1) = 1 and FX(1) = 1 so
that P(X-->Y)(1, 1) = 1 + 1 - 1 = 1 from (6), and F(0, 0) = 0 and
FX(0) = 0 so P(X-->Y)(0, 0) = 1 + 0 - 0 = 1.

Notice also that P(X-->Y)(x, x) = 1 = P(X-->Y)(1, 1) = P(X-->Y)(0, 0)
for any x in the range of X and Y and that P(X-->Y)(1 - x, 1 - x) = 1
for any x in the range of X and Y, so PI exactly has the AO property
of the theorem for (s, s) being the 2-dimensional generalization of s
in the theorem.

Osher Doctorow

From: OsherD on 27 Sep 2007 18:47
>From Osher Doctorow

In my first equation, I meant to type gamma(1 - s), not gamma)(1 - s).

Osher Doctorow

From: Mas Plak on 28 Sep 2007 01:06

"OsherD" <mdoctorow(a)ca.rr.com> wrote in message
news:1190933271.191639.301890(a)22g2000hsm.googlegroups.com...
> >From Osher Doctorow
>
> In my first equation, I meant to type gamma(1 - s), not gamma)(1 - s).
>
> Osher Doctorow
>

you make mistake again.
you get an F


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