On Riesz mean for the coefficients of twisted Rankin-Selberg \(L\)-functions (Q1396402)

Let \(f\) and \(g\) be two normalized Hecke eigen cusp forms of integral weight \(k\) and \(\ell\) (\(k \geq \ell \geq 12)\), respectively, with Fourier coefficients \(a_n\) and \(b_n\). Further let \(\chi\) be a character mod\( d\), and let \[ L_{f\otimes g}(s,\chi) := \sum_{n=1}^\infty c_nn^{-s} \qquad(\Re s >1) \] with \[ c_n := n^{1-(k+\ell)/2}\chi(n)\sum_{m^2|n}a_{n/m^2}b_{n/m^2}m^{k+\ell-2} \] be the Rankin-Selberg series (\(L\)-function) attached to \(f,g\) and \(\chi\). \textit{K. Matsumoto, Y. Tanigawa} and the reviewer [Math. Proc. Camb. Philos. Soc. 127, 117-131 (1999; Zbl 0958.11065)] studied sums of \(c_n\)'s in the case \(f = g\) and \(\chi\) being trivial. By means of Voronoi-type formulas they obtained several results involving the Riesz mean (of order \(\rho\)) \[ \Delta_\rho(x) := {1\over\Gamma(\rho+1)}\sum_{n\leq x}(x-n)^\rho c_n \tag{1} \] when \(\rho=0\) and \(\rho=1\). The author calls this the ``non-twisted case'', and she is interested in the cases \(f\not=g\) or ``\(f = g\) and \(\chi\) is non-trivial'', which she calls the ``twisted cases''. Thus in the non-twisted case \(L_{f\otimes g}(s,\chi)\) has a pole of order one at \(s=1\), while in the twisted case it has no poles. In the present paper it is assumed that \(\chi\) is a primitive character, and it is proved that (see (1)) \[ \Delta_0(x) \ll x^{\alpha/2}d^{\beta/2} \] provided that \(\alpha, \beta\) are real constants such that \[ \Delta_1(x) \ll x^\alpha d^\beta \qquad(\text{ if} d^\beta \leq 2x^{1-\alpha/2}). \] Moreover, if \[ \Delta^*_1(t):= \Delta_1(t) - tL_{f\otimes g}(0,\chi), \] then one has the mean square formula \[ \int_1^X(\Delta^*_1(t))^2 dt = {2d^3\over 13(2\pi)^4}\sum_{n=1}^\infty|c_n|^2n^{-7/4}X^{13/4} + O_\varepsilon(X^{3+\varepsilon} d^{4+\varepsilon}), \] which in the case \(f=g\), \(d=1\) was obtained by Ivić-Matsumoto-Tanigawa (op. cit.). The author uses the methods of this work for her proofs, but there are difficulties coming from the fact that special care must be taken because of the \(d\)-aspect. The technical points involved with this are successfully overcome by the author.