THE IMPACT OF PRIME NUMBER THEORY ON FREQUENCY METROLOGY

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1 THE IMPACT OF PRIME NUMBER THEORY ON FREQUENCY METROLOGY M. PLANAT Laboratoire de Physique et Métrologie des Oscillateurs du CNRS 32 Avenue de l Observatoire, Besanćon Cedex, France planat@lpmo.edu Arithmetic of Beat Frequency Measurements Low frequency noise of electronic oscillators is usefully interpreted in terms ofarithmetic : thisisbecausethemeasurementofthefrequencyf(t)ofan oscillatorundertestcomparestheonef 0 ofareferenceoscillatorthankstoa nonlinear mixing set-up and a filter. The beat frequency f B = p i f 0 q i f(t), withp i andq i integers, () follows from the continued fraction expansion of the frequency ratio ν = f f 0 =[a 0 ;a,a 2,...a i,a,...]=a 0 +/{a +/{a /{a i +/{a...}}}}= p i(a) q pi i(a) q i oftheinputoscillators.herea min a a max,witha min = f0 f cq i, a max = f0 f dq i andf c andf d arethelowandhighfrequencycut-offofthefilter. Sincea intypicalmeasurements,thebeatnoteiswellapproximatedby theconvergent()whichrestrictstothepartialquotienta i intheexpansion. Fig.showsaschematicoftheresultingintermodulationspectrum 2. µ= f B/f0 µ=f /f C 0 3 ν ν p q ν= f /f 0 Figure. The intermodulation spectrum at the output of the mixer+filter set-up. Insomebadcircumstances thepartialquotientsafteradon tplayany ANDREWS2: Frequency Standards and Metrol... on September 3, 200

2 role and frequency jumps occurs randomly at definite values of a leading to a large white frequency noise arising from the detection set-up instead of the oscillator under test. This can be compared to the measurement of time from a moon-sun calendar. Early calendars have been devised from the motion of moon and sun as observed from the earth. The continued fraction expansion oftheratioνbetweenthesunyearandthemoonyearis ν= =[2;2,,2,,,7,...] (2) Thefirstapproximationν=2(with354days)canbecorrectedbyadding onemontheverytwoyears,thesecondone(with369days)maybecorrected byaddingonemontheverythreeyearsandsoon. Fluctuationoftheintegerainthefrequencymeasurementset-uphasthesameaimtocorrectthe measurement versus time. 2 Arithmetic of Phase Locking and /f Frequency Noise Ifonecantrackthefrequencyf(t)bythatf 0 ofavoltagecontrolledoscillator onegetsaphaselockedloop 3. Toafirstapproximationthephaselocking model 2 forthephaseshiftθ n betweenoscillatorsisthearnoldmap θ n+ =θ n +2πΩ csinθ n, (3) whereω= f f 0 isthebarefrequencyratio,c= K f 0 andkistheopenloop gain. Suchanonlinearmapisstudiedbyintroducingthewindingnumber ν=lim n (θ n θ 0 )/(2πn).Thelimitexistseverywhereaslongasc<,the curveνversusωisadevil sstaircasewithstepsattachedtorationalvaluesof Ω= pi q i andwithwidthincreasingwiththecouplingcoefficientc.thephase lockingzonesmayoverlapifc>leadingtochaosfromquasiperiodicity 4. To appreciate the impact of harmonics on the coupling coefficient one can observethateachharmonicofdenominatorq i leadstothesamefluctuating frequencyδf B =q i δf(t).thereareφ(q i )ofthem,whereφ(q i )iseulertotient function,thatisthenumberofintegerslessorequaltoq i andprimetoit;the averagecouplingcoefficientisthusexpectedtobe/φ(q i ) a.wedevelopeda more refined model based on the properties of primes by defining the coupling coefficientasc =cλ(n;q i,p i )with { lnb ifn=b Λ(n;q i,p i )= k,baprimeandn p i mod(q i ) (4) 0 otherwise a Thisshouldbecorrectedtoaccountforthesymmetryofthemixer.Inthebalancedphase bridgeweusedintheexperiments,onlyharmonicswithoddvaluesofp i andq i contribute significantly. ANDREWS2: Frequency Standards and Metrol... on September 3, 2002

3 Thismeansanonzerocouplingtoharmonicsattimesn=p i +q i l,linteger whenevernisapowerofaprime;thecouplingatthefundamentalmodeis theso-calledvonmangoldtfunctionλ(n) 5. AccordingtothegeneralizedRiemannhypothesisonegetstheaverage 6 c av/c= t t n= Λ(n;q i,p i )= +ǫ(t), (5) φ(q i ) withǫ(t)=0(t /2 ln 2 (t))whichisagoodapproximationaslongasq i < t. Abetterestimatemayalsobeobtainedatlargerq i 6. For pi q i = thefluctuatingtermmaybeexpressedintermsofthezerosof theriemannzetafunctionζ(s).thetrivialonesats= 2l,linteger,connect to Bernouilli polynomials; according to Riemann hypothesis(still unproved), theyareinfinitelymanynontrivialzerosatthecriticallines= 2 whichare randomlydistributed.inthegeneralcaseofthe pi q i harmonic,ζ(s)generalizes to a Dirichlet series. One finds numerically that the power spectral density of thefluctuatingtermlookslikea/fnoise. winding number bare frequency ratio Figure2. PhaselockingstepsfortheArnoldmap(plainline),andlackofphaselocking steps for the Arnold von Mangoldt map(dotted line); coupling coefficient c =. We have studied numerically phase locking properties of mapping(3) with c asthecouplingcoefficientinplaceofthestandardonec.themoredrastic effectistopreventthephaselockingoftheoscillators. Fig. 2showsthat thephaselockingstepsareremoved.inadditionthe/fnoiseinthepower spectral density of ǫ(t) converts to /f noise in the fluctuation of the frequency ratioνasshowninfig.3. In conclusion the present theory connects prime numbers to the metrology of time and /f frequency noise in the context ANDREWS2: Frequency Standards and Metrol... on September 3, 2003

4 00 power spectral density Fourier Frequency f Figure 3. /f noise of the winding number in the Arnold von Mangoldt map(dotted line) in comparison to the Arnold map(plain line); coupling coefficient c = 0.2. of phase locking of oscillators. For others links between number theory and physics see mwolf and mwatkins/zeta/physics.htm. 3 Annex:The/fNoiseTermandRiemannHypothesis We remind the relationship between the fluctuating term ǫ(t) and the theory oftheriemannzetafunction 5. Riemann zeta function ζ(s) is defined from the Euler product ζ(s)= n= n s= bprime b s where R(s)>. (6) Riemann s great achievment in 859 was his ability to complete the formula to the whole complex plane of the parameter s. By logarithmic derivation(6) can be rewritten as ζ (s) ζ(s) = Λ(n)n s = n= t s dψ(t)=s t s ψ(t)dt with R(s)>, (7) withthevonmangoldtfunctionλ(n)=ln(b)ifn=b k,baprimeand0 otherwise.functionψ(t)= n t Λ(n)isthesummatoryfunction. ANDREWS2: Frequency Standards and Metrol... on September 3, 2004

5 The inverse transform ψ(t)= 2iπ c+i c i ζ (s) ζ(s) tsds s withc=r(s)>, (8) allowsanestimateofψ(t)ifoneknowsthesingularitiesofζ(s). Thepole of ζ (s) ζ(s) ats=contributest;thepole/sats=0contributes ζ (0) ζ(0) = ln(2π)andthezerosρcontribute tρ ρ.onegets ψ(t)=t(+ǫ(t)) withtǫ(t)= ln(2π) 2 ln( t 2 ) ρ t ρ ρ. (9) Thesecondterminǫ(t)isduetothetrivialzerosofζ(s)whicharelocatedat s= 2l(lapositiveinteger). Thethirdtermisduetotheremainingzeros ofζ(s).billionsofthemhavebeencomputed;allarefoundtobelocatedon thelines= 2.Riemannhypothesisisthe(unsolved)conjecturethatallnon trivial zeros belong to the critical line. These zeros are very irregularly spaced andareresponsiblefortheveryirregularshapeoftheerrortermasshownin Fig.4.Itwasshownnumerically thatthepowerspectraldensityofǫ(t)has 0.04 error term in von Mangoldt program time t Figure 4. The error term ǫ(t) in the summatory function ψ(t). a /f dependance on the Fourier frequency f. Thefluctuatingterm ρ tρ ρ,whereρ= 2 +iy,canbeboundedifone knowsthenumbern(y)ofzerosbetween0andy. SinceN(y) <yln(y), assuming Riemann hypothesis, this implies the von Koch estimate ǫ(t) = O(t /2 ln 2 (t))oftheerrorterm 6. ANDREWS2: Frequency Standards and Metrol... on September 3, 2005

6 4 Annex 2: Harmonic Interactions, /f Noise and the Generalized Riemann Hypothesis Euler s identity(6) can be generalized to the Dirichlet L-series κ(n) L(s,κ)= = R(s)> n s κ(b)b s n= bprime withκ(n)=κ(n)mod(q) for(n,q)= and0 otherwise. (0) In (0)thenotation(n,q) = meansthatnandqarecoprimes b. The Dirichlet character κ(n) is thus a multiplicative function. UsingΛ(n;q,p)inplaceofΛ(n),(7)-(9)canbegeneralizedtothesummatoryfunctionψ(n;q,p)= n t Λ(n;q,p),wheren pmod(q),withthe result ψ(t;q,p)=t(+ǫ(t;q,p)) withtǫ(t;q,p)= L (0,κ) L(0,κ) + t 2m 2m m ρ t ρ ρ. () Theerrorterm isshowninfig. 5. Asaboveitspowerspectral density approximates a /f law. The fluctuating term can be bounded assuming error term in generalized von Mangoldt program time t Figure5. Theerrortermǫ(t;q,p)inthesummatoryfunctionψ(t;q,p);p/q=3/8. the generalized Riemann hypothesis that all non trivial zeros of L-functions belongstothecriticallines= 2. Thesame(poor)estimate6 ǫ(t;q,p)= O(t /2 )ln 2 (t))follows. b Inthissectionweusethenotationp,qinsteadofp i,q i ANDREWS2: Frequency Standards and Metrol... on September 3, 2006

7 5 Annex3: /f Frequency Noise and the Phase Locked Loop From numerous experiments we found(see Fig. 6) that /f frequency noise isapropertyofan(unlocked)phaselockedloop(pll) 3. Itisthusofgreat interesttorelateerrortermsofprimenumbertheorytothestudyofthephase locked loop. If one accounts for the whole set of harmonics, the differential equation forthephaseshiftθ(t;q,p)attheharmonic(p,q)canbewritten 2 θ(t;q,p)+qh(p) r,s K(r,s)sin( s q θ(t;q,p) ω 0t q (qr ps)+θ 0(r,s)) =ω B (p,q), (2) whereω B =2πf B,f B isthebeatfrequencyasgivenin(),k(r,s)isthe effectivegainatharmonic(r,s)andh(p),wherep = d dt,istheopenloop transfer function. Solving(2) is formidable task. It is enough here to observethatthereferencesignalatfrequencyf 0 =2π/ω 0 actsasaperiodic perturbation of the standard model of the PLL. If one neglects harmonic interactions, (2) simplifies to the standard Arnoldmapmodel 2 (3). Phaselockingzonescorrespondstothestairsof thecurveν=lim n (θ n θ 0 )/(2πn)versusΩ,asitisexplainedinSect. 2. TheopenloopgainKinthePLLreflectsintothecouplingcoefficient beat frequency instantaneous time Figure6. Beatfrequency(inHz)closetothephaselockedzone/ofaPLL(reference frequency:f 0 =5MHz).Thepowerspectrumoftheserecordshasapure/fdependance. c=k/f 0. Insuchacaseeithertheloopisphaselockedatsomeharmonicν= p q orνisanirrationalnumber. Quasiperiodicchaosmayoccur onlyatc>. Accountingfortheeffectsofharmonicswecanexpectthat ANDREWS2: Frequency Standards and Metrol... on September 3, 2007

8 thecouplingcoefficientgeneralizesasc =c(+α/φ(q)),αaconstant,since eachharmonicofdenominatorqhasthesameeffectδf B =qδf(t)onthebeat frequencyandtheyareφ(q)ofthem,withφ(q)theeulerfunction.thisleads toalackofphaselockingasintheuncoupledcasec=0;butnochaosis producedasitisthecaseforthelinearmap. Wearethusledtotheunorthodoxconclusionthattheeffectofharmonicsmaybe toproduce adigitalmodulationofthe couplingcoefficient as c av =c(+αω(n;q,p)). Toafirstapproximationtheaveragecouplingcoefficientstillisc,butthereistheerrortermǫ(t;q,p)asgivenin(5). The /f noisewhichispresentinc av movestothewindingnumberνasitis showninfig. 3. Thiswouldexplainthat/FFREQUENCYNOISEISA UNIVERSAL PROPERTY OF UNLOCKED PHASE LOCKED LOOPS as found from experiments. Acknowledgments IthanksmysummerstudentsJ.P.MarilletandE.Henryfortheirhelpinthe experiments leading to the present theory. References. M. Planat, Fluctuation and Noise Letters,, R65 R79(200) 2. M. Planat, Noise, Oscillators and Algebraic Randomness, Lecture Notes in Physics, 550, Springer, Berlin(2000). 3.S.DosSantosandM.Planat,inFractalsandBeyond,Complexityand Fractals in the Sciences, ed. M.N. Novak(World Scientific, Singapore (998)). 4. P. Cvitanovic, in From Number Theory to Physics, ed. M. Waldschmidt et al(springer Verlag, Berlin, 992) 5. H.M. Edwards, Riemann s Zeta Function. Academic Press, N.Y.(974). 6. H. Davenport, Multiplicative Number Theory. Springer Verlag, N.Y. (980) ANDREWS2: Frequency Standards and Metrol... on September 3, 2008