Comments on
G. Pólya's "Bemerkung
uber die integraldarstellung der Riemannschen zeta-funktion"
("Remarks on the integral representation of Riemann's zeta function")
Mark Kac
[from Pólya, G., *Collected Papers, vol.
II: Locations of Zeros *(R.P. Boas, ed.), MIT Press, 1974]
Although this beautiful paper takes one to within
a hair's breadth of Riemann's hypothesis it does not seem to have inspired
much futher work and references to it in the subsequent mathematical literature
are rather scant.
Because of this it may be of interest to relate that
the paper did play a small, but perhaps not wholly negligible, part in
the development of an interesting and important chapter in Statistical
Mechanics.
In the fall of 1951 and the spring of 1952 C.N. Yang
and T.D. Lee were developing their theory of phase transitions which has
since become justly celebrated. To illustrate the theory they introduced
the concept of a "lattice gas" and they were led to a remarkable conjecture
which (not quite in its most general form) can be stated as follows.
**[mathematical text to be inserted - conjecture]**
When I first heard of this conjecture I tried the
simplest case
**[mathematical text to be inserted]**
and somehow Hilfsatz II of Pólya's paper came to
mind.
Here is how, by a slight modification of Pólya's
proof, one can prove the Yang-Lee theorem in the special case.
**[mathematical text to be inserted]**
I showed this proof for the special case to Yang
and Lee. A couple of weeks later they produced their proof of the
general theorem.^{1}I recall Professor Yang telling me at
the time that Hilfsatz II of Pólya, in the form discussed above, was one
essential ingredient in their proof. I have shown Professor Yang
these comments, and I would like to include his recollections here.
"When Lee arrived at Princeton in the fall of 1951,
I was just recovering from my computation of the magnetization of the Ising
model. I realized that the Ising model is equivalent to the concept
of a lattice gas. So, w worked on that and finally produced our paper
I. In the process of doing that, we discovered, by working on a number
of examples, the conjectured unit circle theorem.
"We then formulated a physicist's 'proof' based
on no double roots when the strength of the couplings were varied.
Very soon we recognized that this was incorrect; and for, I would guess,
at least six weeks we were frustrated in trying to prove the conjecture.
I remember our checking into Hardy's book on Inequalities, our talking
to Von Neumann and Selberg. We were, of course, in constant contact
with you all along (and I remember with pleasure your later help in showing
us Wintner's work, which we acknowledged in our paper). Sometime
in early December, I believe, you showed us the proof of the special case
when all the couplings are there and are of equal strength, the case that
you are now writing about in connection with Pólya's collected works.
The proof was fine, but we were still stuck on the general problem.
Then one evening around December 20, working at home, I suddenly recognized
that by making *z*_{1}, *z*_{2}, ... independent
variables and studying their motions relative to the unit circle one could,
through an induction procedure, bring to bear a reasoning similar to the
one used in your argument and produce the complete proof. Once this
idea was there, it took only a few minutes to tighten up all the details
of the argument.
"The next morning I drove Lee to pick up some Christmas
trees, and I told him the proof in the car. Later on, we went to
the Institute; and I remember telling you about the proof at a blackboard.
"I remember these quite distinctly because I'm quite
proud of both the conjecture and the proof. It is not such a great
contribution, but I fondly consider it a minor gem."
T.D. Lee and C.N. Yang,*
Statistical theory of equations of state and phase transitions II. Lattice
gas and Ising model, * Phys. Rev. **87** (1952), pp. 410-419,
especially Appendix II.
**statistical mechanics page
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