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On the parity of the Fourier coefficients of $j$-function


Authors: M. Ram Murty and R. Thangadurai
Journal: Proc. Amer. Math. Soc. 143 (2015), 1391-1395
MSC (2010): Primary 11F30; Secondary 11F03
DOI: https://doi.org/10.1090/S0002-9939-2015-12307-6
Published electronically: January 2, 2015
MathSciNet review: 3314054
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Abstract: Klein’s modular $j$-function is defined to be \[ j(z) = E_4^3/ \Delta (z) = \frac {1}q + 744 + \sum _{n=1}^\infty c(n)q^n\] where $z\in {\Bbb C}$ with $\Im (z) > 0$, $q = \exp (2i\pi z)$, $E_4(z)$ denotes the normalized Eisenstein series of weight $4$ and $\Delta (z)$ is the Ramanujan’s Delta function. In this short note, we show that for each integer $a\geq 1$, the interval $(a, 4a(a+1))$ (respectively, the interval $(16a-1, (4a+1)^2)$) contains an integer $n$ with $n\equiv 7\pmod {8}$ such that $c(n)$ is odd (respectively, $c(n)$ is even).


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Additional Information

M. Ram Murty
Affiliation: Department of Mathematics, Queen’s University, Kingston, Ontario, Canada, K7L 3N6.
MR Author ID: 128555
Email: murty@mast.queensu.ca

R. Thangadurai
Affiliation: Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad 211019, India
Email: thanga@hri.res.in

Keywords: Fourier coefficients, modular forms, $j$-function
Received by editor(s): July 14, 2013
Published electronically: January 2, 2015
Communicated by: Ken Ono
Article copyright: © Copyright 2014 American Mathematical Society