Oscillations of Fourier coefficients of product of \(L\)-functions at integers in a sparse set (Q6594427)

For a normalized Hecke eigen-form \(f\), the Rankin-Selberg \(L\)-function \(R(s,f\times f)\) and \(w\in \mathbb{N}\), define \(\lambda_{w,f\times f}(n)\) by\N\[\N R(s,f\times f)^w=\sum_{n=1}^{\infty}\frac{\lambda_{w,f\times f}(n)}{n^s}, \quad \sigma>1.\tag{17}\N\]\NFor a sparse sequence \(\{x_n\}\) and an arithmetical function \(a(n)\) the power sum \(\sum_{n\le X}a(x_n)^\ell\), \(\ell\in \mathbb{N}\) has been often considered. In the paper, the case \(a(n)=\lambda_{w,f\times f}(n)\) and \(x_n\) is a square-free integer represented as \(\mathcal{Q}(\boldsymbol{x})\), is considered (which is denoted with \(\flat\) on the summation sign), Eqn.(6), where \(\mathcal{Q}(\boldsymbol{x})\) is a primitive positive definite integral binary quadratic form of the fixed discriminant \(D<0\) (corresponding to an imaginary quadratic field). Let \(S_D\) be the set of all inequivalent such forms. The authors consider\N\[\N\begin{aligned} S_1(D,X)&=S(D,X,\ell)=\sideset{}{^\flat}\sum_{\substack{\mathcal{Q}(\boldsymbol{x})\le X\\\N\mathcal{Q}\in S_D, \boldsymbol{x}\in \mathbb{Z}^2}}\lambda_{w,f\times f}(\mathcal{Q}(\boldsymbol{x}))^\ell\\\N&=\mathop{\sum{^\flat}}_{n\le X}\lambda_{w,f\times f}(\mathcal{Q}(\boldsymbol{x}))^\ell W_Dr_D^\ast(n), \end{aligned}\tag{6}\N\]\Nwhere \(r_D^\ast(n)\) arises from the decomposition of the Dedekind zeta-function of the imaginary quadratic field into the product of the Riemann zeta and Dirichlet \(L\)-function attached to it with \(W_D\) being the number of roots of unity. A more complicated related sum is also considered which can be treated verbatim to the above. In previous works, the quadratic form was diagonal (Gaussian field).\N\NFor the proof the generating function\N\[\N L_1(s)=\mathop{\sum{^\flat}}_{n\ge 1}^{}\frac{\lambda_{w,f\times f}(n)^\ell r_D^\ast(n)}{n^s}, \quad \sigma>1\tag{24}\N\]\Nis introduced. The main ingredient is that \(L_1(s)\) can be decomposed into a certain harmless factor \(G_1(s)\) (absolutely convergent product for \(\sigma>\frac{1}{2}\)) and the essential factor \(L_{11}(s)\) consisting of certain powers of the Riemann zeta, Dirichlet \(L\)-function, symmetric power \(L\)-function associated to \(f\), and the twist of the symmetric power \(L\)-function associated to \(f\) by a Dirichlet character \(\chi_D\).\N\NThen the strongest known convexity bounds for \(\frac{1}{2}\le \sigma \le1\) are assembled in Lemmas 3.3--3.5.\N\NThe method depends on the use of the weight function \(w(x)\) which is \(1\) only for \(x\in [2Y,X]\) and \(0\) for \(x<Y\) and \(x>X+Y\) (\(1\le Y\le X/2\)). In the intermediate length \(Y\)-intervals \([Y,2Y]\) and \([X,X+Y]\), \(w(x)\) is a rather smooth function that connects \(1\) with \(0\).\N\N\(\frac{1}{W_D}S_1(D,X)\) is approximated by the series\N\[\N\mathop{\sum{^\flat}}_{n\ge 1}^{}\lambda_{w,f\times f}(n)^\ell r_D^\ast(n)w(n),\N\]\Nwhich is converted into a contour integral along \(\sigma=1+\varepsilon\), \(\varepsilon>0\) by the Mellin inversion of \(w\). Then the procedure is well-known: shifting the path to \(\sigma=\frac{1}{2}+\varepsilon\) passing through the pole at \(s=1\) and then estimating the resulting integral. In the last step of estimation, the convexity bounds are used with \(G_1(s)\) being negligible. Similar argument appears in [\textit{Y. Jiang} and \textit{G. Lü}, J. Number Theory 171, 56--70 (2017; Zbl 1419.11113)] The result is given in Theorem 2.1 in the form\N\[\N S_1(D,X)=XP(\log X)+O\left(X^{1-\frac{1}{\eta}+\varepsilon} \right)\tag{8}\N\]\Nwith \(\eta>0\) given in (9).