HAUPTMODULN AND EVEN-ORDER MOCK THETA FUNCTIONS MODULO 2

초록

The Fourier coefficients c1(n) of the elliptic modular jfunction are always even for n not equivalent to 7 (mod 8). In contrast, for n equivalent to 7 (mod 8), it is conjectured that "half " of the coefficients take odd values. In this article, we first observe in detail when c1(8n-1) is odd and show that the coefficients share the same parity as the coefficients c mu 2 (n) of the 2nd order mock theta function mu 2(q). Furthermore, we prove that this phenomenon also holds among several Hauptmoduln and between Hauptmoduln and even-order mock theta functions.

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