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تسجيل مستخدم جديدOn the distribution of eigenvalues of Maass forms on certain moonshine groups
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In this paper we study, both analytically and numerically, questions involving the distribution of eigenvalues of Maass forms on the moonshine groups $Gamma_0(N)^+$, where $N>1$ is a square-free integer. After we prove that $Gamma_0(N)^+$ has one cusp, we compute the constant term of the associated non-holomorphic Eisenstein series. We then derive an average Weyls law for the distribution of eigenvalues of Maass forms, from which we prove the classical Weyls law as a special case. The groups corresponding to $N=5$ and $N=6$ have the same signature; however, our analysis shows that, asymptotically, there are infinitely more cusp forms for $Gamma_0(5)^+$ than for $Gamma_0(6)^+$. We view this result as being consistent with the Phillips-Sarnak philosophy since we have shown, unconditionally, the existence of two groups which have different Weyls laws. In addition, we employ Hejhals algorithm, together with recently developed refinements from [31], and numerically determine the first $3557$ of $Gamma_0(5)^+$ and the first $12474$ eigenvalues of $Gamma_0(6)^+$. With this information, we empirically verify some conjectured distributional properties of the eigenvalues.
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