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The asymptotic distribution of traces of cycle integrals of the \(j\)-function. (English) Zbl 1287.11060

From the text: We establish an asymptotic formula with a power savings in the error term for traces of cycle integrals of the classical modular \(j\)-function \(j(z)=q^{ - 1}+744+196884q+21493760q^{2}+ ... .\) This implies a conjecture of W. Duke, Ö. Imamoḡlu and Á. Tóth [Ann. Math. (2) 173, No. 2, 947–981 (2011; Zbl 1270.11044)].
Theorem 1.1. For all \(\varepsilon> 0\) we have \[ \frac{\text{Tr}_D(j_m)}{\text{Tr}_D(1)}\rightarrow-24\sigma_1(m)+O_{\varepsilon,m}\left(D^{-\frac{1}{400}+\varepsilon}\right) \] as \(D\to\infty\) through fundamental discriminants. Here \(\sigma_1(m)\) is the divisor function, \(D>0\) a nonsquare.

MSC:

11F30 Fourier coefficients of automorphic forms
11L15 Weyl sums

Citations:

Zbl 1270.11044
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References:

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