Berry Conjecture

The longstanding conjecture that the nonimaginary solutions  of

(1)

where  is the Riemann zeta function, are the eigenvalues of an "appropriate" Hermitian operator . Berry and Keating (1999) further conjecture that this operator is

			(2)
			(3)

where  and  are the position and conjugate momentum operators, respectively, and multiplication is noncommutative. Note that  is symmetric but might have nontrivial deficiency indices, so while physicists define this operator to be Hermitian, mathematicians do not.

REFERENCES:

Berry, M. V. and Keating, J. P. " and the Riemann Zeros." In Supersymmetry and Trace Formulae: Chaos and Disorder(Ed. I. V. Lerner, J. P. Keating, and D. E. Khmelnitskii). New York: Kluwer, pp. 355-367, 1999. http://www.phy.bris.ac.uk/people/berry_mv/the_papers/Berry306.pdf.

Rockmore, D. Stalking the Riemann Hypothesis: The Quest to Find the Hidden Law of Prime Numbers. New York: Vintage, 2006.