From Osher Doctorow
We know that:
1) P(A-->B) = 1 - x + y
and its n-dimensional generalization which replaces real x, y by x-
bar, y-bar (for x = (x1, x2, ..., xn), y = (y1, y2, ..., yn), any
positive integer n, x-bar being the arithmetic mean of the xi
components, etc., is a 1-sided partial inverse of the Euclidean
distance function, which therefore can be called a "proximity
function", at least on the unit interval [0, 1] or the unit square [0,
1] X [0, 1].
It is also obvious that:
2) 1/d is a 2-sided inverse of Euclidean distance d
and so is related to both (1) and to Euclidean distance (inversely to
the latter).
Now consider the Dirichlet series:
3) a1/1^s + a2/2^s + a3/3^s + a4/4^s + a5/5^s + ...
which for ai = 1, i = 1, 2, 3, ... is the Riemann zeta function.
Multiplicatively inverting the denominator of each term therefore
gives "proximity" functions:
4) proximity function i = i^s, i = 1, 2, 3, ....
For s = 1, (4) becomes:
5) proximity function i for s = 1 is 1, 2, 3, 4, 5, ....
which generates again the quantities 2, 4, 5 analyzed in the last few
posts. 1 is generated by 5 - 4, 3 by (5 - 4) + (4 - 2) = 3.
These are all related to automorphic representations and L-functions
of the Langlands Program. See Paul Garrett, Feb. 19, 2005,
http://www.math.umn.edu/~garrett/m/v/l_function.pdf, 10 pages,
"Automorphic representations and L-functions."
Osher Doctorow
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