Parity of the coefficients of Klein’s 𝑗-function

Alfes, Claudia

doi:10.1090/S0002-9939-2012-11502-3

Parity of the coefficients of Klein’s $j$-function
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Proc. Amer. Math. Soc. 141 (2013), 123-130 Request permission

Abstract:

Klein’s $j$-function is one of the most fundamental modular functions in number theory. However, not much is known about the parity of its coefficients. It is believed that the odd coefficients are supported on “one half” of the arithmetic progression $n\equiv 7\pmod {8}$. Following a strategy first employed by Ono for the partition function, we use twisted Borcherds products and results on the nilpotency of the Hecke algebra to obtain new results on the distribution of parity for the coefficients of $j(z)$.

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