Was Riemann the first to consider the 'Hilbert-Pólya idea'?
This is taken from Marcus du Sautoy's book The Music of the Primes (Fourth Estate, 2003) p.286:
"The physicists believe that the reason Riemann's zeros will be in a straight line is that they will turn out to be frequencies of some mathematical drum. A zero off the line would correspond to an imaginary frequency whcih was prohibited by the theory. It was not the first time that such an argument had been used to answer a problem. Keating, Berry and other physicists all learnt. Keating, Berry and other physicists all learnt as students about a classical problem in hydrodynamics whose solution depends on similar reasoning. The problem concerns a spinning ball of fluid held together by the mutual gravitational interactions of the particles inside it. For example, a star is a ball of spinning gas kept together by its own gravity. The question is, what happens to the spinning ball of fluid if you give it a small kick? Will the fluid wobble briefly and remain intact, or will the small kick destroy the ball completely? The answer depends on showing why certain imaginary numbers lie in a straight line. If they do, the spinning ball of fluid will remain intact. The reason why these imaginary numbers do indeed line up is related very closely to the quantum physicists' ideas about proving the Riemann Hypothesis. Who discovered this solution? Who used the mathematics of vibrations to force these imaginary numbers onto a straight line? None other than Bernhard Riemann.
Shortly after his triumph at the Schrödinger Institute, Keating was due to visit Göttingen to lecture on using connections with quantum physics to shed light on the Riemann Hypothesis. Most mathematicians passing through Göttingen take the time to visit the library to examine Riemann's famous unpublished scribblings, his Nachlass. Not only is it a moving experience to feel a bond with such an important figure in the history of mathematics, but the still conatins many unsolved mysteries, locked inside Riemann's illegible scribbles. It has becom the Rosetta Stone of mathematics.
Before Keating set off for Göttingen, one of his colleagues in the mathematics department, Philip Drazin, recommended looking at the part of the Nachlass in which Riemann tackles the classical problem of hydrodynamics. Although Riemann's housekeeper destroyed so many of his papers, the Nachlass still contains a wealth of material, and has been divided into many sections covering different periods of Riemann's life and his many varied interests.
At the library in Göttingen, Keating ordered the two different parts of the Nachlass that he wanted to
consult: one on Riemann's ideas about the zeros in zeta landscape, and the second on his work on
hydrodynamics. When only one pile of papers appeared from the vaults, Keating mentioned that he had asked to
see two parts. Both 'parts' were on the same sheets of paper, the librarian told him. As Keating explored the
pages, he found to his amazement that Riemann had been concocting his proof about rotating balls of fluid at the
very same time that he'd been thinking about the [nontrivial zeros of the zeta function]. The very method by
which modern-day physicists were proposing to force Riemann's zeros to line up had been used by Riemann to
answer the hydrodynamics problem. There, in front of Keating on the same pieces of paper, were Riemann's thoughts
on both problems."
the spectral interpretation
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