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.1I 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 z1, z2, ... 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.

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