Higher power moments of the Riesz mean error term of symmetric square \(L\)-function
Let \[ f(z) = \sum_{n=1}^\infty a_f(n)e^{2\pi iz}\qquad(z = x+iy,\,y>0) \] be the Fourier series attached to a holomorphic cusp form of weight \(k\) for the full modular group \(\mathrm{SL}(2,\mathbb Z)\). Further set \(\lambda_f(n) := a_f(n)n^{(1-k)/2}\), so that \(\lambda_f(n)\) is multiplicative, and for primes \(p\) one has \(\lambda_f(p) = \alpha_f(p) + \beta_f(p)\) with \(\alpha_p(p)\beta_f(p)=1\). The symmetric square \(L\)-function attached to \(f\) is defined, for \(\text{Re}\, s>1\), as \[ \begin{aligned} L(\mathrm{sym}^2f,s)& = \prod_p(1-\alpha_f^2(p)p^{-s})^{-1}(1-p^{-s})^{-1} (1-\overline{\alpha_f^2(p)}p^{-s})^{-1} \cr& = \sum_{n=1}^\infty \lambda_{\mathrm{sym}^2f}(n)n^{-s},\end{aligned} \] and otherwise by analytic continuation. Further set \(c_n := \lambda_{\mathrm{sym}^2f}(n)\), so that \(c_n \ll_\varepsilon n^\varepsilon\), and for \(\rho \geq 0\) consider the Riesz means \[ \frac{1}{\Gamma(\rho+1)}{\sum_{n\leq x}}'(x-n)^\rho c_n = \frac{L(\mathrm{sym}^2f,0)} {\Gamma(\rho+1)}x^\rho + \Delta_\rho(x;\mathrm{sym}^2f), \] where the dash \({}'\) in the sum means that the last summand is halved if \(x\) is an integer, and \(\Delta_\rho(x;\mathrm{sym}^2f)\) is an error term. The authors are concerned with power moments of this function. Their main result is the following: If there exists a real number \(A_0 := A_0(\rho) > 3\) such that \[ \int_1^T |\Delta_\rho(x;\mathrm{sym}^2f)|^{A_0}\,dx \ll_{\rho,\varepsilon} T^{1+\frac{2\rho+1}{3}A_0+\varepsilon} \] then for any integer \(3\leq h < A_0\) we have the asymptotic formula \[ \begin{aligned} \int_1^T \Delta^h_\rho(x;\mathrm{sym}^2f)\,dx &= \frac{6B_\rho(h,c)}{(3+(2\rho+1)h)(2\pi)^{(\rho+1)h}3^{h/2}}T^{1+\frac{2\rho+1}{3}h}\cr& + O\left(T^{1+\frac{2\rho+1}{3}h+\varepsilon}\left(T^{-\delta_\rho(h,A_0)}+T^{-\rho/3}\right)\right), \end{aligned}\leqno(1) \] where the \(O\)-constant depends on \(\rho, h, \varepsilon\). In particular they have asymptotic formulas for the integral in (1) when \(\rho = 1/2\) and \(h = 3,4,5\).
scientific article; zbMATH DE number 5960850