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From: OsherD on 12 Dec 2008 03:00
From Osher Doctorow

A. G. Bagdasaryan of V. A. Trapeznikov Institute for Control Sciences
Russian Academy of Sciences (RAS) Russia in "An elementary and real
approach to values of the Riemann zeta function," arXiv: 0812.1878 v1
[math.NT] 10 Dec 2008, 12 pages, uses a remarkably bold method of
reorganizing the integers to explore the Riemann zeta function
(important in physics) without the usual complex variable machinery
such as analytic continuation and functions of complex variables. The
results are obtained on the negative integers as domain.

The paper is also of extreme interest arguably because the results
express the Riemann zerta function and its alternating series analog
the Dirichlet eta series in terms of Bernoulli numbers Bm,
respectively -Bm+1/(m+1) and (2^(m+1) - 1)Bm/(m+1) where Bm+1 is used
for short to mean B_m+1 (the m+1st Bernoulli number).

If we look look up "Bernoulli numbers" in Wikipedia and Wolfram, then
we obtain:

1) x/(exp(x) - 1) = sum Bn x^n/n!, sum for n = 0 to infinity, for |x|
< 2pi.

Notice the curious fact that exp(x) = sum x^n/n!, so that (1) looks
like an attempt to express x/(exp(x) - 1) as an analog of an
exponential function with coefficients Bn. From the viewpoint of
Probable Causation/Influence (PI), one reason for interest in this
ratio (the ratio of the left side of equation (1)) is that both x and
exp(x) (and so exp(x) - 1) are solutions of the Riccati Differential
Equation without quadratic term, and the latter equation with or
without quadratic term is key in PI. However, exp(x) also turns out
to be key as part of a rational form solution of the subtype of
Riccati Differential Equations that is the Logistic Differential
Equation, and of course it is important in physical Cosmology in
Inflation and elsewhere.

But in a sense, exp(x) - 1 or exp(x) and x are opposite extremes of
solutions of Riccati Differential Equations in terms of slow versus
fast change, and (1) indicates that they are related to each other by
multiplication times a series (the right hand side of (1)) that looks
very much similar to an exponential series. We have already seen
recently in posts here that P(A-->B-->C) has remarkable behavior at
extremes such as P(B) --> 0+ or P(B) --> 1-, and the analogous problem
for exp(x) - 1 or exp(x) versus x seems to be studied with regard to
Bernoulli numbers.

There are even stranger results, including the fact that Bn is the
limit as x --> 0 of the nth derivative with respect to x of x/(exp(x)
- 1) whih can be used to calculate Bn. Notice the appearance of an
extreme (0) limit again.

By the way, the paper of Bagdasaryan is valuable in gathering together
applications of the Riemann zeta function including zeta(3/2) used in
calculating the critical temperature for Bose-Einstein condensate, zeta
(4) in Stefan-Boltzmann law and Wien approximation, and zeta in models
of Quantum Chaos and calculation of Casimir Effect, as well as applied
statistics in Zipf's law, Zipf-Mandelbrot law, physical cosmology.

Osher Doctorow
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