2010

Error Estimates in Horocycle Averages Asymptotics: Challenges from String Theory

Cardella, Matteo A.

Understand

There is an intriguing connection between the dynamics of the horocycle flow in the modular surface $SL_{2}(\pmb{Z}) \backslash SL_{2}(\pmb{R})$ and the Riemann hypothesis.

It appears in the error term for the asymptotic of the horocycle average of a modular function of rapid decay.
We study whether similar results occur for a broader class of modular functions, including functions of polynomial growth, and of exponential growth at the cusp.
Hints on their long horocycle average are derived by translating the horocycle flow dynamical problem in string theory language.

Built on

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H. Furstenberg, The Unique Ergodicity of the Horocycle Flow, Recent Advances in Topological Dynamics, A. Beck (ed.), Springer Verlag Lecture Notes, 318 (1972), 95-115
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P. Sarnak, ”Asymptotic behavior of periodic orbits of the horocycle flow and Eisenstein series.” Comm. Pure Appl. Math. 34 (1981), no. 6, 719–739
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D. Zagier, Eisenstein Series and the Riemann zeta function, Automorphic Forms, Representation Theory and Arithmetic, Studies in Math. Vol. 10, T.I.F.R., Bombay, 1981, pp. 275-301
1981
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Similar

D. Zagier, The Rankin-Selberg method for authomorphic functions which are not of rapid decay in J. Fac. Sci. Tokyo 1981
1981
Cited alongside, same era.
S. G. Dani and J. Smillie, Uniform distribution of horocycle orbits for Fuchsian groups. Duke Math. J. 51 (1984), 185194
1984
Cited alongside, same era.
D. Kutasov and N. Seiberg, “Number Of Degrees Of Freedom, Density Of States And Tachyons In String Theory And Cft,” Nucl. Phys. B 358
1991
Cited alongside, same era.
M. Ratner, Distribution rigidity for unipotent actions on homogeneous spaces. Bull. Amer. Math. Soc. (N.S.) Volume 24, Number 2 (1991), 321-325. M. Ratner, Raghunathan’s topological conjecture and distributions of unipotent flows. Duke Math. J. Volume 63, Number 1 (1991), 235-280
1991
Cited alongside, same era.
S. L. Cacciatori and M. A. Cardella, “Uniformization, Unipotent Flows and the Riemann Hypothesis,” arXiv:1102.1201 [math.NT]
Original
Cited in the paper.
S. L. Cacciatori and M. A. Cardella, “Eluding SUSY at every genus on stable closed string vacua,” arXiv:1102.5276 [hep-th]
Original
Cited in the paper.
D. Zagier, Private communication
Cited in the paper.

Then

A. Verjovsky, ”Arithmetic geometry and dynamics in the unit tangent bundle of the modular orbifold”, in: Dynamical Systems (Santiago 1990), Pitman Res. Notes Math. No. 285 , Longman Sci. Tech., Harlow, 1993, pp. 263?298
1993
Later among the works it cites.
M. Cardella, “A novel method for computing torus amplitudes for ℤ N \mathbb{Z}_{N} orbifolds without the unfolding technique,” JHEP 0905
Original
2009
Later among the works it cites.
S. Cacciatori, M. Cardella, “Equidistribution rates, closed string amplitudes, and the Riemann hypothesis,” JHEP 12 (2010)
Original
2010
Closest in time.
C. Angelantonj, M. Cardella, S. Elitzur and E. Rabinovici, “Vacuum stability, string density of states and the Riemann zeta function,” JHEP 02(2011)
Original
2011
Closest in time.

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