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Order and chaos in some trigonometric series: curious adventures of a statistical mechanic. (English) Zbl 1263.82042

Summary: This paper tells the story of how a MAPLE-assisted quest for an interesting undergraduate problem in trigonometric series led some “amateurs” to the discovery that the one-parameter family of deterministic trigonometric series \[ \mathcal S_p : t \mapsto \sum_{n \in \mathbb N} \mathrm{sin} (n^{-p}t),\;p > 1, \] exhibits both order and apparent chaos, and how this has prompted some professionals to offer their expert insights. As to order, an elementary (undergraduate) proof is given that \[ \mathcal S_p (t) = \alpha_p \operatorname{sign}(t) |t|^{1/p} + O(|t|^{1/(p+1)})\;\forall t\in \mathbb R, \] with explicitly computed constant \(\alpha _{p }\). As to chaos, the seemingly erratic fluctuations about this overall trend are discussed. Experts’ commentaries are reproduced as to why the fluctuations of \[ \mathcal S_p (t) - \alpha_p \operatorname{sign} (t) |t|^{1/p} \] are presumably not Gaussian. Inspired by a central limit type theorem of Marc Kac, a well-motivated conjecture is formulated to the effect that the fluctuations of the \(\lceil t ^{1/(p+1)}\rceil \)-th partial sum of \(\mathcal S_p (t)\), when properly scaled, do converge in distribution to a standard Gaussian when \(t\rightarrow \infty \), though-provided that \(p\) is chosen so that the frequencies \(\{n ^{ - p }\}_{n\in \mathbb N}\) are rationally linear independent; no conjecture has been forthcoming for rationally dependent \(\{n ^{ - p }\}_{n\in \mathbb N}\). Moreover, following other experts’ tip-offs, the interesting relationship of the asymptotics of \(\mathcal S_p (t)\) to properties of the Riemann \(\zeta \) function is exhibited using the Mellin transform.

MSC:

82C40 Kinetic theory of gases in time-dependent statistical mechanics
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
60F05 Central limit and other weak theorems
33B10 Exponential and trigonometric functions
60J25 Continuous-time Markov processes on general state spaces
42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
44A05 General integral transforms

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References:

[1] Ball, M.L.: After 50 years at Courant, Jerry Percus is still ’A little crazy’. Courant Inst. of Math. Sci. Newsletter, Fall/Winter 2008, p. 3
[2] Bleher, P.: On the distribution of the number of lattice points inside a family of convex ovals. Duke Math. J. 67, 461–481 (1992) · Zbl 0762.11031 · doi:10.1215/S0012-7094-92-06718-4
[3] Bleher, P., Cheng, Z., Dyson, F.J., Lebowitz, J.L.: Distribution of the error term for the number of lattice points inside a shifted circle. Commun. Math. Phys. 154, 433–469 (1993) · Zbl 0781.11038 · doi:10.1007/BF02102104
[4] Chamizo, F., Ubis, A.: Some Fourier series with gaps. J. Anal. Math. 101, 179–197 (2007) · Zbl 1211.42003 · doi:10.1007/s11854-007-0007-z
[5] Dyson, F.: Missed opportunities. Bull. Am. Math. Soc. 78, 635–652 (1972) · Zbl 0271.01005 · doi:10.1090/S0002-9904-1972-12971-9
[6] Edwards, H.M.: Riemann’s {\(\zeta\)} Function. Dover, Mineola (1974) · Zbl 0315.10035
[7] Flett, T.M.: On the function $\(\backslash\)sum_{n=1}\^{\(\backslash\)infty}(1/n)\(\backslash\)sin(t/n)$ . J. Lond. Math. Soc. 25, 5–19 (1950) · Zbl 0035.31401 · doi:10.1112/jlms/s1-25.1.5
[8] Graham, S.W., Kolesnik, G.: Van der Corput’s Method of Exponential Sums. London Math. Soc. Lect. Note Ser., vol. 126. Cambridge University Press, Cambridge (1991) · Zbl 0713.11001
[9] Hardy, G.H., Littlewood, J.E.: Notes on the theory of series (XX): on Lambert series. Proc. Lond. Math. Soc. 2(41), 257–270 (1936) · Zbl 0014.30301 · doi:10.1112/plms/s2-41.4.257
[10] Heath-Brown, D.R.: The distribution and moments of the error term in the Dirichlet divisor problem. Acta Arith. 60, 389–414 (1992) · Zbl 0725.11045
[11] Ivić, A.: The Riemann {\(\zeta\)} Function. Dover, Mineola (2003) · Zbl 1114.11071
[12] Iwaniec, H., Kowalski, E.: Analytic Number Theory. Amer. Math. Soc. Colloquium Pub., vol. 53. AMS, Providence (2004) · Zbl 1059.11001
[13] Kac, M.: Sur les fonctions indépendantes V. Stud. Math. 7, 96–100 (1938) · Zbl 0018.07602
[14] Kac, M.: On the distribution of values of trigonometric sums with linearly independent frequencies. Am. J. Math. 65, 609–615 (1943) · Zbl 0061.13710 · doi:10.2307/2371869
[15] Kac, M.: Statistical Independence in Probability, Analysis, and Number Theory. Math. Assoc. of America. Wiley, New York (1959) · Zbl 0088.10303
[16] Kiessling, M.K.-H.: Statistical equilibrium dynamics. In: Campa, A., Giansanti, A., Morigi, G., Sylos Labini, F. (eds.) AIP Conf. Proc., vol. 970, pp. 91–108. Am. Inst. of Phys., New York (2008)
[17] Kowalenko, V., Frankel, N.E., Glasser, M.L., Taucher, T.: Generalised Euler-Jacobi Inversion Formula and Asymptotics Beyond All Orders. London Math. Soc. Lect. Note Ser., vol. 214. Cambridge University Press, Cambridge (1995) · Zbl 0856.33003
[18] McKean, H.P.: Mark Kac: 1914–1984, A Biographical Memoir. Publ. of the Nat. Acad. Sci., Washington (1990)
[19] Miller, S.D., Schmid, W.: The highly oscillatory behavior of automorphic distributions for SL(2). Lett. Math. Phys. 69, 265–286 (2004) · Zbl 1127.11033 · doi:10.1007/s11005-004-0470-8
[20] Montgomery, H.L.: Ten lectures on the interface between analytic number theory and harmonic analysis. CBMS Regional Conf. Series in Math., vol. 84. AMS, Washington (1994) · Zbl 0814.11001
[21] Ninham, B.W., Hughes, B.D., Frankel, N.E., Glasser, M.L.: Möbius, Mellin, and mathematical physics. Physica A 186, 441–481 (1992) · doi:10.1016/0378-4371(92)90210-H
[22] Titchmarsh, E.C.: The Theory of the Riemann Zeta-Function, 2nd edn. Clarendon Press, New York (1986). Edited and with a Preface by D.R. Heath-Brown · Zbl 0601.10026
[23] Reed, M., Simon, B.: Fourier Analysis, Self-Adjointness. Methods of Modern Mathematical Physics, II. Academic Press, San Diego (1975) · Zbl 0308.47002
[24] Riemann, G.F.B.: Gesammelte mathematische Werke und wissenschaftlicher Nachlass. Teubner, Leipzig (1876). Dedekind, R., Weber, H. (eds.) · JFM 24.0021.04
[25] Vinogradov, I.M.: The Method of Trigonometrical Sums in the Theory of Numbers. Dover, Mineola (2004) · Zbl 1093.11001
[26] Zygmund, A.: Trigonometric Series, 3rd edn. Cambridge University Press, Cambridge (2002) · Zbl 1084.42003
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