## The Riemann Zeros and Eigenvalue Asymptotics (1999)

Venue: | SIAM Rev |

Citations: | 40 - 5 self |

### BibTeX

@ARTICLE{Berry99theriemann,

author = {M. V. Berry and J. P. Keating},

title = {The Riemann Zeros and Eigenvalue Asymptotics},

journal = {SIAM Rev},

year = {1999},

volume = {41},

pages = {236--266}

}

### Years of Citing Articles

### OpenURL

### Abstract

Comparison between formulae for the counting functions of the heights t n of the Riemann zeros and of semiclassical quantum eigenvalues En suggests that the t n are eigenvalues of an (unknown) hermitean operator H, obtained by quantizing a classical dynamical system with hamiltonian H cl . Many features of H cl are provided by the analogy; for example, the "Riemann dynamics" should be chaotic and have periodic orbits whose periods are multiples of logarithms of prime numbers. Statistics of the t n have a similar structure to those of the semiclassical En ; in particular, they display random-matrix universality at short range, and nonuniversal behaviour over longer ranges. Very refined features of the statistics of the t n can be computed accurately from formulae with quantum analogues. The Riemann-Siegel formula for the zeta function is described in detail. Its interpretation as a relation between long and short periodic orbits gives further insights into the quantum spectral fluctuations. We speculate that the Riemann dynamics is related to the trajectories generated by the classical hamiltonian H cl = XP. Key words. spectral asymptotics, number theory AMS subject classifications. 11M26, 11M06, 35P20, 35Q40, 41A60, 81Q10, 81Q50 PII. S0036144598347497 1.

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Citation Context ...(4.31) � � Λ −1 p ;Λ −1 p , � �Λ −1 p �ζ (p) � 2 � � exp (iTpξ) � D (0) � � �ζ (p) D (iξ) � � (again ξ = x/�〈d〉). Here γ is the residue of the pole at =-=s =0ofζD(s), 2φ1 is the qhypergeometric function [45],-=- and ζ p D is the pth element of the product over primitive orbits in (3.3). The formal similarity between the results for the Riemann zeros and for the semiclassical eigenvalues is striking, and rei... |

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Citation Context ...orm a group (the inverse multiplications 1/m are missing). Another possibility, closely related to the ideas of [62], is to use not all integers but the group of integers under multiplication (mod k) =-=[68].-=- This would have two advantages. First, it involves only integer dilations. Second, including the characters χ(n) of this group (sets of k complex numbers with unit modulus) opens the possibility of ... |

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Citation Context ...rbits will play a key role.s244 M. V. BERRY AND J. P. KEATING 3. Long Orbits and Universality. In a classically chaotic system, the periodic orbits proliferate exponentially as their period increases =-=[24], with densi-=-ty (3.1) ρ (T) ≡ ∼ number of orbits with periods between T and T + dT dT exp (λT) T as T →∞. Here, λ is the topological entropy of the system. In the cases we are interested in, λ can be i... |

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Citation Context ... roughest level of description, and with the distribution appropriately smoothed, (1.2) π ′ (x) ∼ 1 log x (as implied by the prime number theorem: π(x) ∼ x/log x). One of Riemann’s great ach=-=ievements [7, 8] was to give a-=-n exact formula for π ′ (x), constructed as follows. First, π ′ (x) is expressed in terms of a function J(x) [7, Chap. 1] that has jumps at prime powers: (1.3) π ′ (x)= 1 x ∞� k=1 µkx1/k... |

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Citation Context ...esult of modern mathematical physics, central to the Riemannquantum analogy, is that these isolated periodic trajectories determine the fluctuations in the counting function N(E) of the energy levels =-=[18, 19, 20, 21]. Us-=-ing the notation (2.2), with E replacing t, we can separate N(E) into its smooth and fluctuating parts 〈N(E)〉 and Nfl(E). The averaging is over an energy interval large compared with the mean leve... |

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Citation Context ...eros. A mathematical theory of universal spectral fluctuations already exists in the more conventional context where statistics are defined by averaging over an ensemble. This is random-matrix theory =-=[28, 29, 30, 31, 32]-=-, where the correlations between matrix eigenvalues are calculated by averaging over ensembles of matrices whose elements are randomly distributed, in the limit where the dimension of the matrices ten... |

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Citation Context ...tn (assumed real), the counting function is defined for t>0as (2.1) N (t) ≡ ∞� Θ(t− tn), n=1 where Θ denotes the unit step. Central to our arguments is the fact that N(t) can be decomposed a=-=s follows [11]: (2.2) N (t)=〈N (t)〉 + Nfl (t), wher-=-e � � θ (t) 1 1 1 〈N (t)〉≡ +1= arg Γ + π π 4 2 it � − 1 � t log π +1 2 = t 2π log � � t + 2πe 7 (2.3) � � 1 + O 8 t and Nfl (t)= 1 π lim � � 1 (2.4) Im log ζ + it ... |

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Citation Context ...understanding of the mechanism by which the cancellation occurs [36, 37]; we will discuss it later. Indeed, for the Riemann zeros, (4.12) can be derived [4] using a conjecture of Hardy and Littlewood =-=[38]-=- concerning the pair distribution of the prime numbers. These correlations are important because if the logarithms of the primes (primitive orbit periods) were pairwise uncorrelated, Koff, being the a... |

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Citation Context ...e given by (4.34) Sij = s(xi,xj)= sin{π (xi − xj)} π (xi − xj) (1 − δij) . The analogue of Montgomery’s theorem for the diagonal contributions to ˜ Rn was proved for n = 3 [46] and then fo=-=r all n ≥ 2 [47]. -=-The off-diagonal contributions were calculated using a generalization of the Hardy-Littlewood conjecture for n = 3 and n = 4 [48] and then for all n ≥ 2 [49]. In all cases the results confirm the co... |

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Citation Context ...2 (x) (4.4) of the Riemann zeros tn near n =10 12 , calculated from (4.18) (with τ ∗ =1/4), (2.14), and (2.18) (thin line), compared with � 2 (x) computed from numerically calculated zeros by Odl=-=yzko [39, 40] (thi-=-ck line); all the zeros are close to t = 2.677 × 10 11 , and their smoothed density is 〈d〉 =3.895 .... Note the resurgence resonances (cf. (4.23)) associated with the lowest zeros t1, t2, and t3,... |

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Citation Context ...idual trajectories but on families of trajectories, whose global structure is an important determinant of the energy-level asymptotics. Of interest here is the case where the trajectories are chaotic =-=[15, 16, 17], -=-that is, where E is the only globally conserved quantity and neighbouring trajectories diverge exponentially. Then on a given energy shell (that is, for given E), the usual structure—and the one we ... |

56 |
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Citation Context ...namics have disappeared. Because of the Riemann-quantum analogy, the same behaviour should hold for the pair correlation of the Riemann zeros. Here we make contact with the seminal work of Montgomery =-=[35], who indee-=-d proved (4.9) in that case. Now we observe that in random-matrix theory the exact form factor of the GUE is (4.10) KGUE (τ)=|τ|Θ(1−|τ|)+Θ(|τ|−1). (Θ is the unit step.) For later reference,... |

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Citation Context ...esult of modern mathematical physics, central to the Riemannquantum analogy, is that these isolated periodic trajectories determine the fluctuations in the counting function N(E) of the energy levels =-=[18, 19, 20, 21]. Us-=-ing the notation (2.2), with E replacing t, we can separate N(E) into its smooth and fluctuating parts 〈N(E)〉 and Nfl(E). The averaging is over an energy interval large compared with the mean leve... |

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Citation Context ... and also to high Riemann zeros, in the sense that the spectral or Riemann-zero averages described in the previous paragraph coincide with GUE averages. First, however, we give a very simple argument =-=[33]-=- showing that the approach to universality must be nonuniform. The classical sum rule (3.2) applies to long orbits but not to short ones, because these will reflect the specific dynamics of the system... |

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Citation Context ...esult of modern mathematical physics, central to the Riemannquantum analogy, is that these isolated periodic trajectories determine the fluctuations in the counting function N(E) of the energy levels =-=[18, 19, 20, 21]. Us-=-ing the notation (2.2), with E replacing t, we can separate N(E) into its smooth and fluctuating parts 〈N(E)〉 and Nfl(E). The averaging is over an energy interval large compared with the mean leve... |

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Citation Context ... Remarkably, it is possible to calculate the correction Rc explicitly and in closed form at this level of approximation. The formula was obtained for both the Riemann zeros and for general systems in =-=[41], and independently in [42] f-=-or the Riemann zeros. From (4.18), (4.2), and (2.14), we get (4.20) R c (x) ≈ R 1 c (x)= � ∗ τ −2 dτ cos {2πxτ}, 0 1 2(π 〈d〉) 2 � m,p p m <exp(2π〈d〉τ ∗ ) log 2 p cos pm � ... |

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Citation Context ...idual trajectories but on families of trajectories, whose global structure is an important determinant of the energy-level asymptotics. Of interest here is the case where the trajectories are chaotic =-=[15, 16, 17], -=-that is, where E is the only globally conserved quantity and neighbouring trajectories diverge exponentially. Then on a given energy shell (that is, for given E), the usual structure—and the one we ... |

30 |
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Citation Context ...2 (x) (4.4) of the Riemann zeros tn near n =10 12 , calculated from (4.18) (with τ ∗ =1/4), (2.14), and (2.18) (thin line), compared with � 2 (x) computed from numerically calculated zeros by Odl=-=yzko [39, 40] (thi-=-ck line); all the zeros are close to t = 2.677 × 10 11 , and their smoothed density is 〈d〉 =3.895 .... Note the resurgence resonances (cf. (4.23)) associated with the lowest zeros t1, t2, and t3,... |

30 |
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Citation Context ...ed for n = 3 [46] and then for all n ≥ 2 [47]. The off-diagonal contributions were calculated using a generalization of the Hardy-Littlewood conjecture for n = 3 and n = 4 [48] and then for all n ��=-=� 2 [49]. -=-In all cases the results confirm the conjecture (4.33) and (4.34). The nonuniversal deviations from the GUE formulae (4.33)–(4.34) were calculated for n = 3 and n = 4 [41] using the method outlined ... |

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Citation Context .... Therefore universality for zeros near t should break down beyond about log(t/2π)/ log 2 mean spacings. We regard the observation of the breakdown of random-matrix universality for the Riemann zeros=-= [34]-=-, in accordance with this prediction, as giving powerful support to the analogy with quantum or wave eigenvalues. 4. Periodic-Orbit Theory for Spectral Statistics. In discussing statistics, it will be... |

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Citation Context ...ypothesis is true, all the (infinitely many) tn are real, and are the heights of the zeros above the real s axis. It is known by computation that the first 1,500,000,001 complex zeros lie on the line =-=[9]-=-, as do more than one-third of all of them [10]. Each term in the sum in (1.4) describes an oscillatory contribution to the fluctuations of the density of primes, with larger Retn corresponding to hig... |

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Citation Context ...matical Sciences, Hewlett-Packard, Laboratories Bristol, Filton Road, Stoke Gifford, Bristol BS12 6QZ, United Kingdom (J.P.Keating@bristol.ac.uk). 236sTHE RIEMANN ZEROS AND EIGENVALUE ASYMPTOTICS 237 =-=[1, 2, 3, 4, 5, 6]-=-, but these accounts do not include several recent developments to be described here, especially those in the last part of section 4 and all of sections 5 and 6. To motivate the approach from physics,... |

27 |
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Citation Context ...deriving the universal limiting form of the implied action correlations [37]. (This procedure essentially follows an analogous derivation for the primes themselves, assuming the Montgomery conjecture =-=[44]-=-). An interesting feature of this approach is that it leads to predictions about classical trajectories based on the distribution of quantum energy levels. However, it gives no information about the d... |

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Citation Context ...ntributions to ˜ Rn was proved for n = 3 [46] and then for all n ≥ 2 [47]. The off-diagonal contributions were calculated using a generalization of the Hardy-Littlewood conjecture for n = 3 and n ==-= 4 [48] and-=- then for all n ≥ 2 [49]. In all cases the results confirm the conjecture (4.33) and (4.34). The nonuniversal deviations from the GUE formulae (4.33)–(4.34) were calculated for n = 3 and n = 4 [41... |

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Citation Context ...terms with periods T ∗ + X and T ∗ − X. The resulting “Riemann-Siegel lookalike” formula is ∆(E) ∼ 2B (E) � (5.23) Dn (E) cos {Sn (E)/� − π 〈N (E)〉} + ···. Tn<T ∗ (E) (Fo=-=r a different derivation, see [54]-=-.) With (5.23) it is possible to reproduce some low-lying quantum eigenvalues, and of course the fact that the sum is finite is a major advantage over the infinite divergent series (2.9) and (5.19). H... |

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Citation Context ...netrable walls corresponds to “quantum billiards,” where waves are governed by the Helmholtz equation with Dirichlet boundary conditions, and the (straight) rays are reflected specularly at the wa=-=lls [14]. Of-=- special interest to us is the asymptotics of the eigenvalues En in the semiclassical limit � → 0, which from (2.8) is equivalent to the short-wavelength or high-frequency limit. Waves, in particu... |

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Citation Context ...oliferation in (3.1) cancels the decay of the intensities A 2 j for long orbits. One way to write this is (3.2) 1 lim T→∞ T � A 2 jδ (T − Tj)=1. j This is the sum rule of Hannay and Ozorio de=-= Almeida [25]. -=-Its importance is threefold: first, it does not contain � and so is a classical sum rule. Second, the amplitudes Aj nevertheless have significance in quantum (i.e., wave) asymptotics, because they g... |

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Citation Context ...matical Sciences, Hewlett-Packard, Laboratories Bristol, Filton Road, Stoke Gifford, Bristol BS12 6QZ, United Kingdom (J.P.Keating@bristol.ac.uk). 236sTHE RIEMANN ZEROS AND EIGENVALUE ASYMPTOTICS 237 =-=[1, 2, 3, 4, 5, 6]-=-, but these accounts do not include several recent developments to be described here, especially those in the last part of section 4 and all of sections 5 and 6. To motivate the approach from physics,... |

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Citation Context ...n are real, and are the heights of the zeros above the real s axis. It is known by computation that the first 1,500,000,001 complex zeros lie on the line [9], as do more than one-third of all of them =-=[10]-=-. Each term in the sum in (1.4) describes an oscillatory contribution to the fluctuations of the density of primes, with larger Retn corresponding to higher frequencies.s238 M. V. BERRY AND J. P. KEAT... |

15 |
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Citation Context ...idual trajectories but on families of trajectories, whose global structure is an important determinant of the energy-level asymptotics. Of interest here is the case where the trajectories are chaotic =-=[15, 16, 17], -=-that is, where E is the only globally conserved quantity and neighbouring trajectories diverge exponentially. Then on a given energy shell (that is, for given E), the usual structure—and the one we ... |

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10 |
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Citation Context ...uantum cells (volume hD ) in the volume Ω(E) of the energy surface H = E; thus〈N(E)〉 ≈Ω(E)/hD . For billiards, Ω is proportional to the spatial volume confining the system (this is Weyl’=-=s asymptotics [23]). The mean lev-=-el density is thus (2.19) 〈d (E)〉∼ Ω′ (E) . hD In the quantum formula (2.13), each orbit contributes an oscillation to Nfl(E), with energy “wavelength” (cf. (2.7)) εp = h Tp (E) . (2.20... |

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Citation Context ...ates a discrete spectrum. The analogy is imperfect, because α is continuous, whereas the L-functions cannot be continuously parameterized. A closer analogy is with quantization on a torus phase space=-= [69]-=-, where for topological reasons the permited phases are discrete. The dynamics (6.2) suggests that the system might be closed by connecting the asymptotic positions with the asymptotic momenta. Then p... |

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Citation Context ...s possible to do better, as we shall see later. Now we turn to the quantum analogues of the Riemann-Siegel formula for classically chaotic systems with D>1, as envisaged in [1], explored in detail in =-=[52]-=-, and derived in [53]. These studies are motivated by the hope that such an effective method of computing Riemann zeros might lead to a useful way to calculate quantum eigenvalues. First, the counterp... |

7 |
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Citation Context ...ossible because (5.24) involves the higher transcendental function Erfc, whereas the Riemann-Siegel expansion involves only elementary functions. Several related representations of Z(t) are now known =-=[56, 57, 58]-=-. The improved representation (5.24), together with the explicit correction terms, can readily be adapted to the quantum spectral determinant. The smoothed version of the Riemann-Siegel lookalike (5.2... |

7 |
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Citation Context ...formation, in a phase space. b. The Riemann dynamics is chaotic, that is, unstable and bounded. c. The Riemann dynamics does not have time-reversal symmetry. In addition, we note the recent discovery =-=[60, 61]-=- of modified statistics of the low zeros for the ensemble of Dirichlet L-functions, associated with a symplectic structure. d. The Riemann dynamics is homogeneously unstable. e. The classical periodic... |

6 |
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Citation Context ...is the semiclassical sum rule. Originally [33] the rule was obtained by a different argument, and was mysterious. Now there is a better understanding of the mechanism by which the cancellation occurs =-=[36, 37]-=-; we will discuss it later. Indeed, for the Riemann zeros, (4.12) can be derived [4] using a conjecture of Hardy and Littlewood [38] concerning the pair distribution of the prime numbers. These correl... |

6 |
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Citation Context ...formation, in a phase space. b. The Riemann dynamics is chaotic, that is, unstable and bounded. c. The Riemann dynamics does not have time-reversal symmetry. In addition, we note the recent discovery =-=[60, 61]-=- of modified statistics of the low zeros for the ensemble of Dirichlet L-functions, associated with a symplectic structure. d. The Riemann dynamics is homogeneously unstable. e. The classical periodic... |

5 |
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Citation Context ... of this formula for the hyperbola billiard (a classically chaotic system with D =2) shows that it can reproduce quantum eigenvalues with high accuracy, even resolving near-degenerate pairs of levels =-=[59]. F-=-inally, we note an important clue to the Riemann dynamics, hidden in the asymptotics (5.10), (5.11) of the Riemann-Siegel expansion (5.6). It concerns the implied small exponential exp{−πt} (cf. th... |

5 |
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Citation Context ...y rotated version of the inverted harmonic oscillator P 2 − X 2 , which in turn is a complexified version of the usual harmonic oscillator P 2 + X 2 . Some of these connections have been noted befor=-=e [63, 64, 65, 66, 67]. -=-The first-order operator XP is the simplest representative of this class, with the monomials (6.5) avoiding the complications of the parabolic cylinder eigenfunctions of P 2 − X 2 .s262 M. V. BERRY ... |

4 |
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