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やはり英語版が詳しいね
https://en.wikipedia.org/wiki/Hilbert%E2%80%93P%C3%B3lya_conjecture
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History
In a letter to Andrew Odlyzko, dated January 3, 1982, George Polya said that while he was in Gottingen around 1912 to 1914 he was asked by Edmund Landau for a physical reason that the Riemann hypothesis should be true, and suggested that this would be the case if the imaginary parts t of the zeros

1/2 +it

of the Riemann zeta function corresponded to eigenvalues of an unbounded self-adjoint operator.[1] The earliest published statement of the conjecture seems to be in Montgomery (1973).[1][2]

David Hilbert did not work in the central areas of analytic number theory, but his name has become known for the Hilbert?Polya conjecture for reasons that are anecdotal.

References
1 Odlyzko, Andrew, Correspondence about the origins of the Hilbert?Polya Conjecture.
http://www.dtc.umn.edu/~odlyzko/polya/index.html
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.
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