黎曼零点谱系传出天外来音，无人识谱

    进入本世纪，黎曼Zeta零点谱系音乐大行其道，红红火火。

    黎曼Zeta函数的零点谱系，整整齐齐，排列在一条垂直线上，零点的间隔不一，如同乐谱。

    利用电声合成器，从中可以播放出天外来音，但是，无人识谱。

    请读者参阅附件。

袁萌  陈启清 2月22日

附件：黎曼零点谱系的音乐功能 （原文）                   

FAST TRACK COMMUNICATION

Hearing the music of the primes: auditory complementarity and the siren song of zeta

M V Berry

H H Wills Physics Laboratory, Tyndall Avenue, Bristol BS8 1TL, UK

E-mail: asymptotico@physics.bristol.ac.uk

Received 2 July 2012 Published 31 August 2012 Online at stacks.iop.org/JPhysA/45/382001

Abstract A counting function for the primes can be rendered as a sound signal whose harmonies, spanning the gamut of musical notes, are the Riemann zeros. But the individual primes cannot be discriminated as singularities in this ‘music’, because the intervals between them are too short. Conversely, if the prime singularities are detected as a series of clicks, the Riemann zeros correspond to frequencies too low to be heard. The sound generated by the Riemann zeta function itself is very different: a rising siren howl, which can be understood in detail from the Riemann–Siegel formula.

PACS numbers: 02.10.De, 02.30.Nw, 43.60.ac, 43.66.Hg

S Onlinesupplementarydataavailablefromstacks.iop.org/JPhysA/45/382001/mmedia

1. Introduction

Riemann [1, 2] showed that the uctuations of the prime numbers about their mean density can be described by a Fourier-like series of oscillations, whose frequencies are given by the celebrated complex zeros of his zeta function. The implied analogy with music has often been noted in lectures, in the title of a popular book [3], and by Bombieri [4]: ‘To me, that the distribution of prime numbers can be so accurately represented in a harmonicanalysisisabsolutelyamazingandincrediblybeautiful.Ittellsofanarcane music and a secret harmony composed by the prime numbers.’ As described later, the ‘music’ is easy to create on a computer; it can be heard online [5], accompanying visual depictions of the underlying signal. My purpose here is to explore this idea further, to see how far the primes, and the associated Riemann zeta function, can be represented by a sound signal incorporating the gamut of musical notes. The simplest prime signal is described in section 2. The implications for hearing the music are analyzed in section 3. The main result is that there is a sense in

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J. Phys. A: Math. Theor. 45 (2012) 382001 Fast Track Communication

whichtheFourierandprimerepresentationsarecomplementary:whenthemusicoftheprimes is synthesized as a superposition of Riemann zeros, harmonies can be heard but individual primes are inaudible; and if the signal is slowed so as to hear the primes, the harmonies are belowthethresholdofhearing.Thesignalhasfractalaspects,discussedinsection4.Thevery different sound generated by the Riemann zeta function is analyzed in section 5.

2. Prime counting signal Thesimplestprimecountingfunctionmightseemtobethestaircase:π(x)=numberofprimes less than x. But it is well known [1] that the connection with the Riemann zeros is simpler with the counting function for prime powers pn, using the convenient weighting logp, namely Riemann’s psi function ψ(x) = pn

ψuct(x) =−

∞ n=1

xρ ρ

---

ρ = 1 2 ±itn, n=1,2,3...

---

. (2.4) The numbers tn are the heights of the zeros; if the Riemann hypothesis (RH) is true, all the complex zeros lie on the critical line Reρ =1/2, so all tn are real. AssumingRH,eachtermin(2.4)representsanoscillationwhosephaseisarg(xρ)=tnlogx. The corresponding angular frequency d(phase)/dx=tn/x depends on x. This is awkward for a soundsignal,inwhichitisdesirableforeachRiemannzerotorepresentapuretone.Therefore we change to a time variable τ proportional to logx: x=exp(aτ), (2.5) involving a scaling constant a. It is convenient to remove the factor √x in ψuct, associated with Reρ =1/2, and thus dene the sound signal representing the primes as S(τ) =exp−1 2aτψuct(exp(aτ))=−2Re ∞ n=1 exp(2πiνnτ) 1 2 +itn . (2.6) Assuming RH, this function has a discrete spectrum, whose frequencies—the harmonies of the primes—are νn = atn 2π . (2.7) The amplitudes are 1/t2 n + 1 4. If RH would be false, some of the νn would be complex, and the spectrum would not be purely discrete. Extending the denition of ‘music’ to include any signal with a discrete spectrum, this enables a nontechnical statement of RH [6]: the primes contain music.

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3. Complementarity of primes and Riemann zeros

To implement S(τ)asmusic,wechoose thefrequency ν1,associated withthelowest Riemann zero, as the lowest note on the piano keyboard, and replace the sum (2.6) by the truncated version SN(τ) =−2Re N n=1 exp(2πiνnτ) 1 2 +itn , (3.1) including N zeros, where tN is associated with the highest note on the piano keyboard. Thus, with τ measured in seconds, ν1 =27.5Hz(musicalnoteA0) νN =4186.01Hz(musicalnoteC8). (3.2) The scaling constant a in (2.5), and the index N, now follow from (2.7): a=12.224, tN = νN ν1 t1 =2151.57⇒N =1657. (3.3) This gives the ‘Riemann scale’ as the set of frequencies ν1, ν2,...,νN. It is easy to create a computer program to enable the scale (2.7) and the ‘music’ (3.1) to be heard. One such program accompanies this paper as supplementary material (available fromhttp://stacks.iop.org/JPhysA/45/382001/mmedia),alongwithsoundclipsofthescaleand signal.Aversionincorporating100zeroshasbeenpostedonline[5].Interpretingthesoundas music requires some imagination: although the low zeros can be discerned as ghostly growls, the signal sounds like noise, for reasons explained in section 4. Alternative implementations are easy to explore. For example, starting at a higher note makes the low harmonies stand out more clearly, at the price having fewer notes in the musical gamut: for ν1 = A2 = 110 Hz, corresponding to a=48.896, N=297. From its denition in terms of ψ(x), the signal (2.6) has singularities: discontinuities corresponding to the prime powers pn, at timesτ =(n/a)logp. These singularities cannot be heard in the prime music as dened here. To understand why, note rst that for very short times it is possible to discriminate individual prime powers in the truncated series (3.1), even with far fewer zeros N, as illustrated in gure 1(a) for 0<τ<0.2 s. But this association soon dissolves, as illustrated in gures 1(b)–(d). The reason is that to resolve detail in ψ(x) orS(τ) on a scale x = logx corresponding to the spacing between primes at x, the synthesis must include at least the rst M Riemann zeros, given by the phase change (tM logx) =tM x x = tM logx x =2π,i.e.tM =2π x logx . (3.4) It now follows from the known counting function of the zeros [1], namely N(t) ≈ (t/2π)log(t/2πe), that M ∼x=exp(aτ). (3.5) This implies that discrimination of the jumps at individual prime powers, even for times inaudibly close to the start of the signal, would require an unfeasibly large number of Riemann zeros: for τ = 1 s,M∼204000; and for τ = 10 s, M∼1.2 × 1053. With M = N = 1657 zeros, corresponding to the highest piano note, primes cease to be discriminated for τ>log(a)/N∼0.6 s. Increasing the gamut to include the full range of human hearing (optimistically, up to 20 kHz) hardly helps. EvenifthenumberofRiemannzerosisincreasedvastlybeyondtheaudiblerange,sothat individual primes could be discriminated in principle, they would still be inaudible, because

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0.00 0.05 0.10 0.15 0.20 -0.4

0.0

0.4

1.00 1.02 1.04 1.06 1.08 1.10

-0.4

0.0

0.4

9.90 9.92 9.94 9.96 9.98 10.0

-0.4

0.0

0.4

0 2 4 6 8 10

-0.5

0.0

0.5

τ(s)

(a)( b)

(c)( d)

2 3

4

5 7

98

11

Figure 1. (a) Comparison of the truncated signal S50(τ) (equation (3.1)) (dotted curve) with the exact S(τ) (equation (2.6)) (full curve), for 0 <τ<0.2 s; the numbers at the jumps indicate the corresponding prime powers in ψ(x). (b) Truncated signal S1657(τ), for 1.0 s <τ<1.1 s. (c) Truncated signal S1657(τ), for 9.9 s <τ<10.0 s. (d) Truncated signal S1657(τ), for 0 <τ<10 s.

the time intervals τ between their singularities in the signal are too short to be heard. For the primes at time τ,

τ =

x ax =

logx ax = τ exp(−aτ). (3.6) Even for the rather short time τ = 1 s,τ ∼5 μs; and for τ = 10 s, τ ∼8 × 10−53 s. As gures 1(b)–(d) illustrate, at such times the singularities at individual prime powers are invisible. Instead, the graphs look fractal, an aspect to be discussed in the next section. Ofcourse,itispossibletoheartheindividualprimepowersingularitiesinS(τ)asaseries of clicks, simply by slowing the signal, that is, by reducing the scaling a. For example, to hear the prime powers 2, 3, 4, 5, 7, 8, 9 in the time interval 0 <τ<10 s, it is necessary to take a= log(10)/10 = 0.2306. But then the lowest Riemann zero t1 contributes with frequency ν1 = 0.517 Hz—far below the threshold of hearing—and the lowest piano note frequency A0 = 25.5 Hz is reached only at the zero t413. So, it is possible to hear the prime power clicks, or the Riemann zero harmonies, but not bothatthesametime.Thiscomplementarityreectsthedifferentscalesatwhichtheindividual Riemann zeros and the individual prime powers occur in the signal S(τ). Each Riemann zero describes an oscillation with period Tn which is independent of τ: Tn = 1/νn = 2π/atn (cf (2.7)). The prime powers—jumps in S(τ)—correspond to the much smaller exponential scale(3.6).TheperiodTn representsascaleofoscillatoryclusteringofprimepowersinwhich each cluster contains Tn/τ = (2π/atn)exp(aτ)/τ prime powers; the number increases exponentially as τ increases.

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On the spectral interpretation of the Riemann zeros as energy levels (eigenvalues) of a classicallychaoticdynamicalsystem[6–8],theprimepowersrepresentrepetitionsofperiodic orbits. S(τ) in ( 2.6) is a trace formula, with singularities at the prime powers (periodic orbits) represented as oscillations determined by the Riemann zeros (energy levels); this is complementary to the spectral density trace formula [9–11] in which singularities at the energylevels(Riemannzeros)arerepresentedasoscillationsdeterminedbytheperiodicorbits (primes). Discriminating individual primes in the signal S(τ) synthesized from the Riemann zerosisanalogousto‘inversequantumchaology’[12],inwhichtheperiodsofclassicalperiodic orbits are determined from the quantum energy levels. It would be interesting to know if the auditory complementarity identied here extends to ‘music’ representing classically chaotic dynamical systems, in which the harmonies are the quantum energy levels.

4. Fractal structure of the prime signal

On the nest scale (3.6), exponentially small as τ increases, S(τ) consists of singularities at prime powers, illustrated in gure 1(a). On coarser scales, the graph of S(τ) looks fractal (gures 1(b)–(d)). The associated dimension D can be estimated as follows. Writing (2.6) as S(τ) =constant×Ren exp(iωnτ) 1 4 +iωn/a (4.1) with ωn =tn/a, the corresponding power spectrum is P(ω) ∝n δ(ω−ωn) ω2 n + 1 4a2 ∼ 1 ω2 dn(ω) dω = 1 ω2 logaω 2π , (4.2) where the last equality incorporates the asymptotic density of the tn. NowweusetheresultthatthegraphofaFourierseriesS(τ)withuncorrelatedphasesand power-law power spectrum P(ω) ∝ ω−μ with 1 <μ<3 is a continuous but nondifferentiable curve with fractal dimension D = (5−μ)/2 [13, 14]. Up to logarithms, (4.2) is a power-law with μ = 2, giving D = 3/2. The observation that this is the dimension of the graph of Brownian motion in one space dimension quanties the pseudo-randomness of the prime powers on coarse scales and explains why the music sounds like noise. Visually, comparison of gures 1(b) and (c) with other fractal curves [14] with a variety of dimensions is consistent withthevalueD=3/2.Amorerenedmeasure-theoreticanalysis,involvingconceptsbeyond the fractal dimension and incorporating the logarithm, would probably indicate additional weak scale-dependent roughness in the graph of S(τ).

5. The song of zeta The natural way to render the zeta function ζ(s) on the critical line s=1/2+it as a sound is by the scaled version Z(at) of the real function [1] Z(t) =exp(iθ( t))ζ1 2 +it, (5.1)in which the phase θ(t) is θ( t) =Imlog
1 4 + 1 2it− 1 2t logπ. (5.2) The sound generated by Z(at), as described in the supplementary material (available from http://stacks.iop.org/JPhysA/45/382001/mmedia), is very different from the music of the primes, in ways that depend on the scaling a. Fora = 1000, it resembles the rising note of a siren; for a=2500, it is a banshee howl; and for a=5395 (a choice explained later) it is an unnerving scream.

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These sounds can be understood by representing Z(t) as a series of oscillations. To sufcient accuracy this given by the ‘main sum’ of the Riemann–Siegel expansion [1, 15]—a version of the Dirichlet series incorporating the functional equation for ζ(s):

ZRS(t) =2

√t/2π n=1

cos{θ( t)−t logn} √n . (5.3) Becauseoftheoor(integerpart)function√t/2π ,thisisanitesuminwhichthenumberof termsincreasesslowlywitht.Thecorrespondingweakdiscontinuitiesarebarelyaudiblewhen ZRS(t)isrenderedasasound(andinanycasecouldbeeliminatedbyincludingRiemann–Siegel correction terms [15] or by smoothing the discontinuities [16]). The oscillations in (5.3) involveθ(t), whose large-t asymptotic form is

θ( t) ≈

1 2

t log

---

 t 2πe

---

− 1 8

π. (5.4) Thus the instantaneous frequencies of the oscillations with indices n are νn(t) = 1 2π d(phase) dt ≈ 1 2π log

1 n
 t 2π
. (5.5) These can be regarded as rising tones, quasi-monochromatic because the frequencies scarcely vary over an oscillation period: νn(t +1/νn(t))−νn(t) νn(t) ≈ 1 νn(t)2 dνn(t) dt = π t log21 nt 2π  

1 (5 .6) forallrelevanttandn.Thelogarithmicspectrum(5.5)contrastswiththeexponentialspectrum νn = ν12n/12 of the semitones of the musical scale. InthescaledversionZRS(at)ofthesum(5.3),thehighestfrequencycorrespondston=1: νmax(t) = alogat 2π4 π . (5.7) Forasoundplayedforthetimeinterval0ttmax,acanbedeterminedtocorrespondtoany choice of the highest frequency νmax(tmax) in this zeta music. For tmax =20 s, and the highest note C8 = 4186 Hz on the piano keyboard (equation (3.2)), this gives a = 5395; a = 2500 corresponds to νmax(20)=1787 Hz, and a=1000 corresponds to νmax(20)=642 Hz. The very different sound of the zeta music Z(t) as compared with the prime music S(τ) (equation (2.6)) is reected in the power spectrum. In terms of the angular frequency ω = 2πν, inverting (5.5) gives n(ω) =
 t 2π exp(−ω). (5.8) Thus the power spectrum corresponding to (5.3) is P(ω) = 1 n(ω)

 

 

 

 

 

 

dn(ω) dω

 

 

 

 

 

 

=

log
 t 2π −ω
. (5.9) Ignoring the step function

, this is a at spectrum, corresponding to white noise. This was notedinapreviousstudy[17,pp253–260],emphasizingtheverydifferentspectraofζ(σ +it) on and off the critical line σ =1/2. However, the cutoff is important because it indicates that the zeta music is band-limited. Over sufciently long times, the cutoff frequency νmax in (5.7) will rise above the audible range. Even then, the logarithmic distribution of frequencies (5.5) sounds different from simulated white noise in which the frequencies (√t/2π of them) are uniformly distributed throughout the range 0 <ν<νmax. And of course the at band-limited spectrum of zeta is very different from the fractal spectrum of the prime music as discussed in section 4.

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References

[1] Edwards H M 2001 Riemann’s Zeta Function (Mineola, NY: Dover)

[2] Borwein P, Choi S, Rooney B and Weirathmueller A 2008 The Riemann Hypothesis: A Resource for the Acionado and Virtuoso Alike (Berlin: Springer)

[3] du Sautoy M 2003 The Music of the Primes (New York: Harper Collins)

[4] BombieriE 1992 Primeterritory:exploringtheinnitelandscapeatthebaseofthenumbersystemTheSciences 32 30–6

[5] Stopple J 2004 Riemann zeta function and explicit formula (http://www.math.ucsb.edu/∼stopple/index.html)

[6] Berry M V and Keating J P 1999 The Riemann zeros and eigenvalue asymptotics SIAM Rev. 41 236–66

[7] BerryM V1986Riemann’szetafunction:amodelforquantumchaos?QuantumChaosandStatisticalNuclear Physics (Lecture Notes in Physics vol 263) ed T H Seligman and H Nishioka (New York: Springer) pp 1–17

[8] Berry M V 2008 Three quantum obsessions Nonlinearity 21 T19–26 [9] Gutzwiller M C 1971 Periodic orbits and classical quantization conditions J. Math. Phys. 12 343–58

[10] Balian R and Bloch C 1972 Distribution of eigenfrequencies for the wave equation in a nite domain: III. Eigenfrequency density oscillations Ann. Phys., NY 69 76–160

[11] Selberg A 1956 Harmonic analysis and discontinuous groups in weakly Riemannian spaces with applications to Dirichlet series J. Indian Math. Soc. 20 47–87

[12] Aurich R and Steiner F 2001 Orbit sum rules for the quantum wave functions of the strongly chaotic Hadamard billiard in arbitrary dimensions Found. Phys. 31 569–92 [13] Mandelbrot B B 1982 The Fractal Geometry of Nature (San Francisco, CA: Freeman)

[14] Berry M V and Lewis Z V 1980 On the Weierstrass–Mandelbrot fractal function Proc. R. Soc. A 370 459–84

[15] Berry M V 1995 The Riemann-Siegel formula for the zeta function: high orders and remainders Proc. R. Soc. Lond. 450 439–62

[16] Berry M V and Keating J P 1992 A new approximation for zeta(1/2 +it) and quantum spectral determinants Proc. R. Soc. Lond. 437 151–73

[17] Crandall R E 1996 Topics in Advanced Scientic Computation (New York: Springer)

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