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Zwegers showed that a mock theta function can be completed to form essentially a real analytic modular form of weight $1/2$ by adding a period integral of a certain weight $3/2$ unary theta series. This theta series is related to the holomorphic modular form called the shadow of the mock theta function. In this paper, we discuss the computation of shadows of the second order mock theta functions and show that they share the same shadow with a mock theta function which appears in the Mathieu moonshine phenomenon.
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참고문헌 (23)
참고문헌 신청
A. Dabholkar / / Quantum Black Holes, Wall Crossing, and Mock Modular Forms / arXiv:1208.4074 [hep-th]
A. Folsom / 2016 / Quantum mock modular forms arising from eta-theta fnctions / Res. Number Theory 2 : 41 ~
B. Gordon / 2012 / A survey of classical mock theta functions, Partitions, q-Series, and Modular Forms / Springer : 95 ~ 144
D. Zagier / 2007 / Ramanujan's mock theta functions and their applications / Seminaire Bourbaki 60eme annee (986)
G. E. Andrews / 1981 / Mordell integrals and Ramanujan's "lost" notebook, Analytic number theory (Philadelphia, PA., 1980) / Lecture Notes in Math. 899 : 10 ~ 18
A. Folsom / 2016 / Quantum mock modular forms arising from eta-theta fnctions / Res. Number Theory 2 : 41 ~
B. Gordon / 2012 / A survey of classical mock theta functions, Partitions, q-Series, and Modular Forms / Springer : 95 ~ 144
D. Zagier / 2007 / Ramanujan's mock theta functions and their applications / Seminaire Bourbaki 60eme annee (986)
G. E. Andrews / 1981 / Mordell integrals and Ramanujan's "lost" notebook, Analytic number theory (Philadelphia, PA., 1980) / Lecture Notes in Math. 899 : 10 ~ 18
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