Natural Science Forum / Physics / General Physics / January 2008

Einstein : a small error of calculation...

In a communication on the quantum theory of monoatomic
ideal gases made in 1924 to the academy of sciences of Prussia:

Preussische Akademie der Wissenschaft
(1924, SB Phys.-math., S. 261-267)
Quantentheorie des einatomigen idealen Gases

Einstein uses this approximate formula:

z(y) = y - 0.1768*y^2 - 0.0034*y^3 - 0.0005*y^4

in order to eliminate 'x' from the system:

y = sum(x^k/k^(3/2), {k, 1, oo})
z = sum(x^k/k^(5/2), {k, 1, oo})

This approximate formula turns out to be mistaken,
the coefficient of y^4 being five times too large.

The approximate formula correctly calculated
(with computer and "mathematica") is indeed this one:

zc(y) = y - 0.1768*y^2 - 0.0033*y^3 - 0.0001177*y^4

Checking can be done with z(zeta(3/2)), which must
theoretically output zeta(5/2) = 1.34149 :

z(zeta(3/2)) - > 1.3219

compare to:

zc(zeta(3/2)) - > 1.34149

Anyone can be mistaken, even the great Einstein...

V.Astanoff

p.s. a colleague of mine makes me observe that, in 1924,
it was rather difficult to use a computer!