Conjugacy classes of and spectral rigidity

Author:
Ralph Phillips

Journal:
Math. Comp. **64** (1995), 1287-1306, S35

MSC:
Primary 11F72; Secondary 11Y70

MathSciNet review:
1303088

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Abstract: The free group is generated by and , and setting defines a unitary character on for . A program is devised to compute

*x*, and the remainder can be written as . The values of are computed for all traces between 3202 and 4802 (here ). The 's cluster around 0.6, attaining a maximum of . Finally, it is proved that the remainder has a negative bias by showing that the mean normalized remainder converges to a negative limit.

**[1]**R. Aurich, E. B. Bogomolny, and F. Steiner,*Periodic orbits on the regular hyperbolic octagon*, Phys. D**48**(1991), no. 1, 91–101. MR**1098656**, 10.1016/0167-2789(91)90053-C**[2]**M. V. Berry,*Semiclassical theory of spectral rigidity*, Proc. Roy. Soc. London Ser. A**400**(1985), no. 1819, 229–251. MR**805089****[3]**Dennis A. Hejhal,*The Selberg trace formula for 𝑃𝑆𝐿(2,𝑅). Vol. I*, Lecture Notes in Mathematics, Vol. 548, Springer-Verlag, Berlin-New York, 1976. MR**0439755****[4]**Henryk Iwaniec,*Prime geodesic theorem*, J. Reine Angew. Math.**349**(1984), 136–159. MR**743969**, 10.1515/crll.1984.349.136**[5]**W. Luo and P. Sarnak,*Number variance for arithmetic hyperbolic surfaces*, Comm. Math. Phys.**161**(1994), no. 2, 419–432. MR**1266491****[6]**Ralph Phillips and Zeév Rudnick,*The circle problem in the hyperbolic plane*, J. Funct. Anal.**121**(1994), no. 1, 78–116. MR**1270589**, 10.1006/jfan.1994.1045**[7]**R. Phillips and P. Sarnak,*The spectrum of Fermat curves*, Geom. Funct. Anal.**1**(1991), no. 1, 80–146. MR**1091611**, 10.1007/BF01895418**[8]**R. Phillips and P. Sarnak,*Cusp forms for character varieties*, Geom. Funct. Anal.**4**(1994), no. 1, 93–118. MR**1254311**, 10.1007/BF01898362**[9]**Charles Schmit,*Quantum and classical properties of some billiards on the hyperbolic plane*, Chaos et physique quantique (Les Houches, 1989) North-Holland, Amsterdam, 1991, pp. 331–370. MR**1188422****[10]**Atle Selberg,*Collected papers. Vol. I*, Springer-Verlag, Berlin, 1989. With a foreword by K. Chandrasekharan. MR**1117906****[11]**Scott A. Wolpert,*Disappearance of cusp forms in special families*, Ann. of Math. (2)**139**(1994), no. 2, 239–291. MR**1274093**, 10.2307/2946582

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DOI:
https://doi.org/10.1090/S0025-5718-1995-1303088-5

Article copyright:
© Copyright 1995
American Mathematical Society