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Ramanujan introduced mock theta functions and described them by an "order" which he did not define. As a result of the work of Zwegers and others we now have a better understanding of mock theta functions. They appear as the holomorphic projection of weight 1/2 harmonic Maass forms and in the theta expansions of meromorphic Jacobi forms. Given this modern understanding one wonders if there is a natural definition of the "order" which agrees with Ramanujan's. On the Wikipedia page on mock modular forms one finds the statement "Ramanujan's notion of order later turned out to correspond to the conductor of the Nebentypus character of the weight 1/2 harmonic Maass forms which admit Ramanujan's mock theta functions as their holomorphic projections." I can check that this is true in a few specific cases (e.g. the order 3 mock theta functions studied by Bringmann and Ono) but have not been able to find this statement in the literature, hence my questions. First, does the definition of the order in the Wikipedia article agree with the orders (2,3,5,6,7,8,10) of the mock theta functions given later in the same article? Second, is there a reference to the literature for this definition?

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I'm afraid there isn't very much in the literature about the order. If you really want to know, you'll have to check it yourself... I think Ramanujan actually only defined the order for the functions of order 3, 5 and 7 (in his last letter to Hardy). Mock theta functions of different order show up in his Lost Notebook, but if I´m not mistaken, he doesn´t label those. By various other people they were later called of order 2,6,8,10,etc. It could very well be that those were not labeled according to Ramanujan's "definition" and/or the modern definition.

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Thanks for the response and welcome to MO. What a curious thing this order is with Ramanujan never defining it and the modern definition appearing only on Wikipedia. –  Jeff Harvey Sep 9 '11 at 18:27

This is not a characterization of the order: it is a definition of the order. You cant really ask if it is the same as Ramanujan's definition of the order, because Ramanujan never defined the order in general. All he did was say what the order was in a few examples, so you can't really do any more than check it is the same on these examples.

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Point taken. I have edited my question accordingly to ask whether the order as defined by the Wikipedia article agrees with the orders used to label the mock theta functions later in the same article and to ask if this definition appears in the literature rather than just on the Wikipedia page. –  Jeff Harvey Sep 8 '11 at 17:55

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