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.

  • 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.

    Up [ Return to home page ]