From Osher Doctorow
There are so many applications of the Riemann zeta function to physics
that it is difficult to keep track of them, though admittedly most of
them involve open questions as to what the Causation is or else simply
state and derive and describe the applications.
"Riemann zeta" as keywords yield 93 papers in arXiv, but since they
are very closely related to other zeta functions, the keyword "zeta"
on arXiv yields 563 papers.
Perhaps the deepest relationship to physics in the literature other
than to Probable Causation/Influence (PI) as described in the previous
posting is in "Bernoulli numbers and solitons" by M-P. Grosset and A.
P. Veselov of Lougborough U. Leicester U.K. in arXiv: math.GM/0503175
v1 9 Mar 2005, 5 pages, together with "Generalization of a relation
between the Riemann zeta function and Bernoulli numbers," by S. C.
Woon of Trinity College Cambridge U. U.K. arXiv: math.NT/9812143 v1 24
Dec 1998 12 pages.
Basically, Bernoulli polynomials and numbers and hence the Riemann
zeta function are related to the soliton solutions of the KdV
Equation, especially the 1-soliton solution:
1) u = -2 sech^2(x - 4t), u(x, 0) = -2 sech^2(x)
with regard to which Grosset and Veselov obtain the new equation:
2) B_2n = [(-1)^(n-1)/2^(2m + 1)] I(Dx^(m-1)(sech^2(x))^2 dx
where I is the integral from -infinity to +infinity, and B_2n is the
2nth even Bernoulli number.
ArXiv has 33 papers on Bernoulli numbers, and I haven't checked how
many they have on Bernoulli polynomials but presumably even more.
This is all also related to the Faulhaber polynomials via D. B.
Fairlie and A. P. Veselov's "Faulhaber and Bernoulli polynomials and
solitons," Physica D 152-153 (2001) 47-50.
The Riemann zeta function z is well known to relate to Bernoulli
polynomials B by:
3) z(2n) = -(1/2)(2pi)^(2n)(-1)^n B_2n /(2n!)
which Woon generalizes, where n is any positive integer.
Osher Doctorow
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