Natural Science Forum / Physics / General Physics / July 2008

From Osher Doctorow

Although the Fundamental Set of Probable Causation/Influence (PI):

1) {0, 1, 2, 3, 4, 5, 7, 10, 11, 13, 26}

contains the numbers 10 and 11 from Superstring and Supersymmetry and
related physics, the number 10 doesn't show up as much as 0, 1, 2, 3,
4, 5 or even 7 as fundamental in papers outside those physics topics,
but Bakir Farhi of U. du Maine Le Mans France, in "A curious result
related to Kempner's series," arXiv: 0807.3518 v1 [math.NT] 22 Jul
2008, proves the remarkable results:

2) lim (sum 1/k) = 10 log 10 where sum is over k = 1 to infinity
provided that positive integer k contains digit 9 in decimal
representation exactly r times, and limit is taken as r --> infinity.

If this seems strange, then look at this:

3) lim (sum 1/k) = 10 log 10 for the same scenario as above (2) except
9 is replaced by any one digit in the set {0, 1, 2, 3, 4, 5, 6, 7, 8,
9}.

Farhi has 8 papers in arXiv.

The series sum 1/k without restrictions beyond to positive integers k
is of considerable importance throughout mathematics and physics, so
the result is not as esoteric as it may seem.  Also, although it is
true that the unlimited sum 1/k for k positive integers is divergent,
it was proven in A. J. Kempner's work that it is convergent when
restricted to positive integers k whose decimal representations do not
contain any digit 9, and this was extended to series where the decimal
rperesentation contains only a limited number of a particular nonempty
set of digits by Irwin and others.

If nothing else, this result makes 10 "fundamental" in mathematics or
at least number theory in the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
beyond its usual function of being a base for the decimal system.

Osher Doctorow