Symmetry of the Riemann operator

Ž . Chaos quantization conditions, which relate the eigenvalues of a Hermitian operator the Riemann operator with the non-trivial zeros of the Riemann zeta function are considered, and their geometrical interpretation is discussed. q 1999 Published by Elsevier Science B.V. All rights reserved. PACS: 11.25.M; 05.45.M; 02.10.L For a long time a challenge for mathematical physics has been and is still the idea, due to Hilbert at the dawn of the quantum age, to relate the nontrivial zeros of the Riemann zeta function with a spectrum of a Hermitian operator in a Hilbert space. Ž . The Riemann zeta function z s is defined for complex sssq ir and Res)1 by the equation ` 1 z s s . 1 Ž . Ž . Ý s n 1 w x The Riemann hypothesis 1 states that the non-trivŽ . ial complex zeros of z s lie on the critical line Ress1r2. This is a very good motivation for understanding the original Riemann conjecture to find a self-adjoint Ž Ž .. operator E or a positive one E 1yE defined in a Hilbert space and with a spectrum given by the non-trivial zeros of the zeta function. In favour of w x this idea are the rigorous results 2,3 , on the spectral theory of the Laplace–Beltrami operator on the Poincare complex upper half-plane, showing that the ́ eigenfunctions are automorphic with respect to a discrete group of fraction-linear transformations. The most important application of representation theory w x to automorphic forms is the work of Selberg 4 who has shown that, if the fundamental domain of the Ž . discrete subgroup G;SL 2, R is compact, then the spectrum of the elliptic operator for which the eigenvalue equation E 2 E 2 1 2 2 yy q fs 1qr f 2 Ž . Ž . 4 2 2 ž / E x E y has G-invariant solutions is determined by the real Ž . values r given by z 1r2q irr2 s0, where n G Ž . z s is a meromorphic function of s on the whole G plane. The spectrum is continuous if the fundamental domain has a finite volume. w x There has been theoretical 5,13 , and experimenw x tal 6,7 , evidence in support of the Riemann hypothesis, based on the fluctuations of spacing between consecutive zeros of zeta. w x In a recent paper, Connes 8 has given a spectral interpretation of the critical zeros and a geometrical 0370-2693r99r$ see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. Ž . PII: S0370-2693 99 00172-0 ( ) B. AneÕarPhysics Letters B 45