1549380232-Automorphic_Forms_and_Applications__Sarnak_

336 AUDREY TERRAS, ARITHMETICAL QUANTUM CHAOS

Figure 1. The Tree of Quantum Chaos

is t he Sinai billiard which is a square with a circular hole cut out center d at the
center of the squ are. Column ( e) is from th positive imaginary parts of zero's
of t he Riemann zeta function (from the 1551th to the 160 0t h zero). Colu mn (f)
i equ ally spaced -the picket fence or uniform distribution. Column (g) comes
from Sarnak [ 67 ] and corresponds to eigenvalues of the P oincare Lap lacia n on the
fundamental domain of t he modular group SL(2, Z) consisting of 2 x 2 integer
matrices of determinant 1. From t he point of view of randomness, columns (g) and