The distributions of coefficients of triple product \(L\)-functions over arithmetic progressions (Q6612911)

Let \(f\), \(g\) and \(h\) be three distinct normalized primitive holomorphic cusp forms of even integral weights \(k_1\), \(k_2\) and \(k_3\) for the full modular group \(\mathrm{SL}(2, \mathbb{Z})\), respectively. Let \(\lambda_{f \times f \times f}(n)\), \(\lambda_{f \times f \times g}(n) \) and \(\lambda_{f \times g \times h}(n)\) denote the \(n\)th coefficients of triple product \(L\)-functions \(L(f \times f \times f, s)\) and \(L(f \times f \times g, s)\) associated with \(f\) and \(f\), \(g\) and \(L(f \times g \times h, s)\) associated to \(f\), \(g\), \(h\), respectively. Let \(q\) be a prime with \((q, l)=1\). The purpose of this paper is to show the following asymptotic formulas \N\[ \sum_{\substack{n \leqslant x \\ n \equiv l \pmod q}} \lambda_{f \times f \times f}^2 (n)= \frac{C_f x P_4(\log x)}{\varphi(q)}+ O_{f, \varepsilon} \left(qx^{1-\frac{3}{2}\eta_1+\varepsilon}\right)\]\N\[ \sum_{\substack{n \leqslant x \\ n \equiv l \pmod q}} \lambda_{f \times f \times g}^2(n)=\frac{C_{f,g } x P_1(\log x)}{\varphi(q)}+ O_{f, g, \varepsilon} \left(qx^{1-\frac{3}{2}\eta_2+\varepsilon}\right)\]\N\[ \sum_{\substack{n \leqslant x \\ n \equiv l \pmod q}} \lambda_{f}^2(n) \lambda_{g}^2(n) \lambda_{h}^2(n)=\frac{C_{f,g, h } x }{\varphi(q)}+ O_{f, g, h, \varepsilon} \left(q^{22/33 +\varepsilon}x^{22/33 +\varepsilon}\right),\] \Nfor any \(\varepsilon > 0\), where \(C_f , C_{f,g}, C_{f, g, h} > 0\) are some suitable constants, and \(P_r(t)\) denotes the polynomial of \(t\) with degree \(r\), and \(\eta_1 = 46/2127\) and \(\eta_2 =23/1085\).