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Andrew Odlyzko: Correspondence about the origins of the Hilbert-Polya Conjecture

The Hilbert-Polya Conjecture says that the Riemann Hypothesis is true because non-trivial zeros of the zeta function correspond (in a certain canonical way) to the eigenvalues of some positive operator. This conjecture is often regarded as the most promising way to prove the Riemann Hypothesis. Very little is known about its origins. Mathematical folk wisdom has usually attributed its formulation to Hilbert and Polya, independently, some time in the 1910s. However, there appears to be no published mention of it before Hugh Montgomery's 1973 paper on the pair correlation of zeros of the zeta function.

Enclosed here are copies of some letters that attempted to trace the history of the Hilbert-Polya Conjecture. The first letter from Polya appears to present the only documented evidence about the origins of the conjecture.

Correspondence with George Polya: The two letters by Polya (1887-1985) were written when he was 94. According to N. G. de Bruijn, at that stage in his life, Polya usually dictated letters to his wife, and only signed them himself. The fact that he wrote both letters out in his own handwriting suggests he was very interested in the subject. The account of the formulation of the conjecture in the first letter is consistent with what Polya had told Dennis Hejhal in a personal conversation.

Andrew Odlyzko to George Polya, December 8, 1981:
Dear Professor Polya:
I have heard on several occasions that you and Hilbert had independently conjectured that the zeros of the Riemann zeta function correspond to the eigenvalues of a self-adjoint hermitian operator. Could you provide me with any references? Could you also tell me when this conjecture was made, and what was your reasoning behind this conjecture at that time?
The reason for my questions is that I am planning to write a survey paper on the distribution of zeros of the zeta function. In addition to some theoretical results, I have performed extensive computations of zeros of the zeta function, comparing their distribution to that of random hermitian matrices, which have been studied very seriously by physicists. If a hermitian operator associated to the zeta function exists, then in some respects we might expect it to behave like a random hermitian operator, which in turn ought to resemble a random hermitian matrix. I have discovered that the distribution of zeros of the zeta function does indeed resemble the distribution of eigenvalues of random hermitian matrices of unitary type.
Any information or comments you might care to provide would be greatly appreciated.
Sincerely yours,
Andrew Odlyzko
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George Polya to Andrew Odlyzko, January 3, 1982:
Dear Mr. Odlyzko:
Many thanks for your letter of December 8. I can only tell you what happened to me.
I spent two years in Goettingen ending around the begin of 1914. I tried to learn analytic number theory from Landau. He asked me one day: "You know some physics. Do you know a physical reason that the Riemann hypothesis should be true." This would be the case, I answered, if the nontrivial zeros of the Xi-function were so connected with the physical problem that the Riemann hypothesis would be equivalent to the fact that all the eigenvalues of the physical problem are real.
I never published this remark, but somehow it became known and it is still remembered.
With best regards.
Your sincerely,
George Polya
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Andrew Odlyzko to George Polya, January 18, 1982:
Dear Professor Polya:
Thank you very much for your letter of January 3 and the information about the origins of your conjecture about zeros of the zeta function. As you may know, the physicists have extensively studied the distribution of eigenvalues of random hermitian matrices. Now the idea is that if there is an operator associated to the zeta function, its eigenvalues might in some respects behave like those of a random hermitian matrix. This chain of reasoning is, of course, very weak; but surprisingly enough, it seems to work as is shown by the enclosed graph, and other results that I have obtained. In any case, I will send you copies of my papers on this subject for comment as soon as they are ready.
Sincerely yours,
Andrew M. Odlyzko
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George Polya to Andrew Odlyzko, April 26, 1982:
Dear Dr. Odlyzko:
Please, excuse the delay of this answer to your January letter. I am sick since almost two years, I was unable to read. A few days ago I acquired a reading machine which enabled me to read your letter. I do not understand yet the graphs but I am awaiting your announced papers.
Sincerely yours,
G. Polya
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Correspondence with Olga Taussky-Todd: Since David Hilbert (1862-1943) could not be consulted, I tried the only person I knew who had worked extensively with Hilbert, namely Olga Taussky-Todd. Unfortunately, she did not have any information on the subject.

Andrew Odlyzko to Olga Taussky-Todd, January 19, 1982:
Dear Professor Taussky-Todd:
Unfortunately, I did not have much of a chance to talk to you at the Santa Barbara meeting in October; and, therefore, I would like to ask for your assistance through this letter.
I am trying to write up some of my results on the distribution of zeros of the zeta function; and in order to make the paper complete, I would like to find out about the origin of the famous conjecture, usually ascribed to Hilbert and Polya, that the zeros of the zeta function are associated to eigenvalues of a self-adjoint hermitian operator. As far as I know, this conjecture was published neither by Hilbert nor by Polya. I recently wrote to Polya, and received in return a description of how he came to make this conjecture. However, I know nothing about Hilbert's reasoning. In your work with him, did he ever mention this conjecture to you? Any information you could supply would be greatly appreciate.
Sincerely yours,
Andrew M. Odlyzko
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Olga Taussky-Todd to Andrew Odlyzko, January 25, 1982:
Dear Dr. Odlyzko:
It is nice to hear from you. I too am sorry not to have had some conversation with you in Santa Barbara. I am also sorry that I am rather ignorant about the question you are asking. In fact I would appreciate to see Polya's reply to you. I never had any conversations with Hilbert on number theory. At the time of my work on his papers in Goettingen he had no interest in number theory, only in logic.
However, I think that H. Kisilewsky, now at Concordia University in Montreal, talked to me about the fact you are mentioning, and I think that he connected it with A. Weil's ideas. Maybe you ought to write to him. He would appreciate hearing from you anyhow. He does, however, usually not reply immediately. I would be grateful to share further information about this with you.
With best wishes,
Olga Taussky-Todd
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