Petersson scalar products and \(L\)-functions arising from modular forms (Q1987606)

The Rankin-Selberg convolution and the related \(L\)-functions are widely used in the study of number theory. In this paper, the author considers the Rankin-Selberg convolution of holomorphic modular forms of the same level. More precisely, let \(f(z)=\sum_{n=0}^\infty a_n e^{2\pi \sqrt{-1} z}\) and \(g(z)=\sum_{n=0}^\infty b_n e^{2\pi \sqrt{-1} z}\) be two modular forms of level \(\Gamma_0(N)\). Denote the weights of \(f(z)\) and \(g(z)\) by \(l\) and \(l'\), respectively (assuming \(l\geq l'\)). Consider the Rankin-Selberg \(L\)-function \[L(s, f, g):= \sum_{n=1}^\infty \frac{a_n \overline{b}_n}{n^s}. \quad (\Re (s)\gg 0)\] The author shows that the above \(L\)-function has a meromorphic continuation to the whole \(s\)-plane, and has a functional equation under \(s\mapsto -(l-l')+1-s\). Moreover, if \(l=l'\), and \(f(z)\), \(g(z)\) have the same character, the author obtains a formula \[\langle f ,g\rangle_{\Gamma_0(N)}=c\cdot \mathrm{Res}_{s=l} L(s,f,g)\] relating the Petersson product of modular forms and the residue of corresponding Rankin-Selebrg \(L\)-function, with an explicit constant \(c\). Such a formula generalizes the work of \textit{H. Petersson} [Comment. Math. Helv. 22, 168--199 (1949; Zbl 0032.20601)]. And if either the weights or the characters of \(f\) and \(g\) are distinct, the author also obtains a special value formula for \(L(1-l, f, g)\). As an application, the above results give some estimations of the sum of Fourier coefficients \[\sum_{0<n\leq X} a_n \overline{b}_n\] as \(X\rightarrow \infty\). In particular, since the cases of half integral weights are also included, the main results of this paper can be applied to the study of quadratic forms. For example, let \(k>0\), and \(r_k(n)\) be the number of representations of an positive integer \(n\) as a sum of \(k\) squares, the author shows that \[\begin{aligned} &\quad\ \sum_{0<n \leq X} r_k(n)r_{k-4m} (n)\\ &\sim \frac{(1-(1-(-1)^m) 2^{-k+2m+1})\pi^{k-2m}\zeta(k-2m-1)}{\Gamma(k/2-2m)\Gamma(k/2) \zeta(k-2m)(1-2^{-k+2m})}\cdot\frac{X^{k-2m-1}}{k-2m-1}, \end{aligned}\] which generalizes a result of \textit{W. Müller} [J. Number Theory 51, No. 1, 48--86 (1995; Zbl 0840.11020)]. Here \(m\in \mathbb{Z}_{\geq 0}\) with \(k>\max\{4m,2\}\). Moreover, an estimation for \(\sum_{0<n\leq X} r_k(n)r_{k-2m}(n)\) (as \(X\rightarrow \infty\)) is also obtained by considering modular forms of level \(\Gamma_0(4)\).