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Berry Conjecture


The longstanding conjecture that the nonimaginary solutions E_n of

 zeta(1/2+iE_n)=0,
(1)

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

H^~=1/2(xp+px)
(2)
=-i(xd/(dx)+1/2),
(3)

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


See also

Riemann Hypothesis, Riemann Zeta Function

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References

Berry, M. V. and Keating, J. P. "H=xp 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.

Referenced on Wolfram|Alpha

Berry Conjecture

Cite this as:

Weisstein, Eric W. "Berry Conjecture." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BerryConjecture.html

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