Zeroes of zeta functions and symmetry.

*(English)*Zbl 0921.11047Ten years ago, A. Odlyzko made what is considered by many specialists to be perhaps the most striking discovery – at the phenomenological level – about the zeta function since the work of Riemann. He found that the local spacings of the zeros obey the laws for the scaled spacings between the eigenvalues of a typical large unitary matrix, namely the laws of the Gaussian Unitary Ensemble (GUE). An impressive plot of the spacings \(\delta_j\), for \(10^{20}\leq j\leq 10^{20}+ 7\cdot 10^6\), fitting exactly the Gaudin distribution is the first figure in the paper, where a total of 11 plots are depicted.

The main aim of the authors is to contribute to the investigations that have been carried out by a number of groups during the last 20 years, trying to elucidate the suggestion by Hilbert and Polya, that there may exist a natural interpretation of the zeros of the Riemann zeta-function. The evidence is becoming quite convincing nowadays. Aside from the facts mentioned in the paragraph above, concerning the local spacing distributions of the high zeros of the Riemann zeta function (what can also be extended to many generalizations of it), it turns out that the low-lying zeros of various families of zeta functions follow laws corresponding to the eigenvalue distributions of members of the classical groups. Another point concerns the case of zeta functions of curves over finite fields and their generalizations, where a spectral interpretation of their zeros exists in terms of eigenvalues of Frobenius on cohomology.

In the paper, all these developments are reviewed, albeit not in a chronological order of discovery. As the authors themselves warn, they only concentrate on the line of work sketched above, ignoring the standard body of important work on zeta functions and \(L\)-functions that has been done with other purposes. The paper has 26 pages. The first section is devoted to the Montgomery-Odlyzko law, providing different plots for the distribution of nearest-neighbor spacings for the zeros of the Riemann zeta-function (already mentioned above), for the Ramanujan \(L\)-function and for the \(L\)-function associated to \(E\), and a plot of the pair correlation for the zeros of zeta based on \(8\cdot 10^6\) zeros near the \(10^{20}\)-th zero, versus the GUE conjectured density. Section 2 is devoted to the uses in this context of the random matrix models, which were introduced by Wigner in the 50’s, in an attempt at describing the resonance line of heavy nuclei. In section 3 they carry the analysis to the zeta functions corresponding to function fields, as introduced by Artin, and devote section 4 to the low lying zeros. Phenomenologically, it is found that the distribution of the low-lying zeros of certain families follow the laws dictated by symmetries associated with the family. Some applications are described in section 5, followed by a concluding section 6.

The final comment in the paper expresses the authors’ belief that for each \(L\)-function of a family, \(L(s,f)\), \(f\in{\mathcal F}\), there is a natural interpretation of the zeros of \(L(s,f)\) as the eigenvalues of an operator \(U(f)\) on some space \(H\). As \(f\) varies over \({\mathcal F}\) the \(U(f)\)’s would become equi-distributed in the space of such operators with a given symmetry type. In particular, the Riemann zeta-function sits in a family that has a symplectic symmetry, and thus the corresponding operator should preserve a symplectic form.

The main aim of the authors is to contribute to the investigations that have been carried out by a number of groups during the last 20 years, trying to elucidate the suggestion by Hilbert and Polya, that there may exist a natural interpretation of the zeros of the Riemann zeta-function. The evidence is becoming quite convincing nowadays. Aside from the facts mentioned in the paragraph above, concerning the local spacing distributions of the high zeros of the Riemann zeta function (what can also be extended to many generalizations of it), it turns out that the low-lying zeros of various families of zeta functions follow laws corresponding to the eigenvalue distributions of members of the classical groups. Another point concerns the case of zeta functions of curves over finite fields and their generalizations, where a spectral interpretation of their zeros exists in terms of eigenvalues of Frobenius on cohomology.

In the paper, all these developments are reviewed, albeit not in a chronological order of discovery. As the authors themselves warn, they only concentrate on the line of work sketched above, ignoring the standard body of important work on zeta functions and \(L\)-functions that has been done with other purposes. The paper has 26 pages. The first section is devoted to the Montgomery-Odlyzko law, providing different plots for the distribution of nearest-neighbor spacings for the zeros of the Riemann zeta-function (already mentioned above), for the Ramanujan \(L\)-function and for the \(L\)-function associated to \(E\), and a plot of the pair correlation for the zeros of zeta based on \(8\cdot 10^6\) zeros near the \(10^{20}\)-th zero, versus the GUE conjectured density. Section 2 is devoted to the uses in this context of the random matrix models, which were introduced by Wigner in the 50’s, in an attempt at describing the resonance line of heavy nuclei. In section 3 they carry the analysis to the zeta functions corresponding to function fields, as introduced by Artin, and devote section 4 to the low lying zeros. Phenomenologically, it is found that the distribution of the low-lying zeros of certain families follow the laws dictated by symmetries associated with the family. Some applications are described in section 5, followed by a concluding section 6.

The final comment in the paper expresses the authors’ belief that for each \(L\)-function of a family, \(L(s,f)\), \(f\in{\mathcal F}\), there is a natural interpretation of the zeros of \(L(s,f)\) as the eigenvalues of an operator \(U(f)\) on some space \(H\). As \(f\) varies over \({\mathcal F}\) the \(U(f)\)’s would become equi-distributed in the space of such operators with a given symmetry type. In particular, the Riemann zeta-function sits in a family that has a symplectic symmetry, and thus the corresponding operator should preserve a symplectic form.

Reviewer: E.Elizalde (Barcelona)

##### MSC:

11M06 | \(\zeta (s)\) and \(L(s, \chi)\) |

14G10 | Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) |

11M41 | Other Dirichlet series and zeta functions |

81Q99 | General mathematical topics and methods in quantum theory |

##### Keywords:

spacing distributions of zeros; zeros of the Riemann zeta-function; zeta functions of curves over finite fields; Montgomery-Odlyzko law; Ramanujan \(L\)-function; pair correlation; random matrix models; symplectic symmetry
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\textit{N. M. Katz} and \textit{P. Sarnak}, Bull. Am. Math. Soc., New Ser. 36, No. 1, 1--26 (1999; Zbl 0921.11047)

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