Duality involving the mock theta function f(q)
نویسندگان
چکیده
We show that the coefficients of Ramanujan’s mock theta function f(q) are the first non-trivial coefficients of a canonical sequence of modular forms. This fact follows from a duality which equates coefficients of the holomorphic projections of certain weight 1/2 Maass forms with coefficients of certain weight 3/2 modular forms. This work depends on the theory of Poincaré series, and a modification of an argument of Goldfeld and Sarnak on Kloosterman–Selberg zeta functions.
منابع مشابه
Ramanujan’s Radial Limits
Ramanujan’s famous deathbed letter to G. H. Hardy concerns the asymptotic properties of modular forms and his so-called mock theta functions. For his mock theta function f(q), he asserts, as q approaches an even order 2k root of unity, that we have f(q)− (−1)(1− q)(1− q)(1− q) · · · ` 1− 2q + 2q − · · · ́ = O(1). We give two proofs of this claim by offering exact formulas for these limiting valu...
متن کاملIDENTITIES AND CONGRUENCES FOR RAMANUJAN’S ω(q)
Recently, the authors [3] constructed generalized Borcherds products where modular forms are given as infinite products arising from weight 1/2 harmonic Maass forms. Here we illustrate the utility of these results in the special case of Ramanujan’s mock theta function ω(q). We obtain identities and congruences modulo 512 involving the coefficients of ω(q).
متن کاملOn flushed partitions and concave compositions
Abstract. In this work, we give combinatorial proofs for generating functions of two problems, i.e. flushed partitions and concave compositions of even length. We also give combinatorial interpretation of one problem posed by Sylvester involving flushed partitions and then prove it. For these purposes, we first describe an involution and use it to prove core identities. Using this involution wi...
متن کاملPARTIAL THETA FUNCTIONS AND MOCK MODULAR FORMS AS q-HYPERGEOMETRIC SERIES
Ramanujan studied the analytic properties of many q-hypergeometric series. Of those, mock theta functions have been particularly intriguing, and by work of Zwegers, we now know how these curious q-series fit into the theory of automorphic forms. The analytic theory of partial theta functions however, which have q-expansions resembling modular theta functions, is not well understood. Here we con...
متن کاملJACOBI’S TRIPLE PRODUCT, MOCK THETA FUNCTIONS, AND THE q-BRACKET
In Ramanujan’s final letter to Hardy, he wrote of a strange new class of infinite series he called “mock theta functions”. It turns out all of Ramanujan’s mock theta functions are essentially specializations of a so-called universal mock theta function g3(z, q) of Gordon–McIntosh. Here we show that g3 arises naturally from the reciprocal of the classical Jacobi triple product—and is intimately ...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2007