On the fourth moment in the Rankin-Selberg problem (Q948898)

Let \(c_n\) be the Dirichlet coefficients of the normalized Rankin-Selberg convolution of a cusp form \(\varphi\) of a given weight for the full modular group and let \(\Delta(x,\varphi)=\Delta(x)\) denote the error term in the asymptotic formula for \(\sum_{n\leq x}c_n\). In this paper the author continues his study on mean values of \(\Delta(x)\). In earlier work, the author obtained the mean square estimate \[ \int_0^X \Delta^2(x)\,dx\ll_\varepsilon X^{1+2\beta+\varepsilon}, \] where \(\beta \) depends on the Lindelöf function. In the present paper, he proves an estimate for the fourth moment \[ \int_0^X\Delta^4(x)dx\ll_\varepsilon X^{3+\varepsilon}, \] which would follow directly from the above mean square estimate only under the Lindelöf hypothesis. The tools used to get the fourth moments are the classical Rankin-Selberg estimate \(\Delta(x)\ll x^{\frac 35}\), truncated Voronoï\ type formulae and \textit{P. Shiu}'s (see the final remark) Brun-Titchmarsh type result on a class of multiplicative functions in short intervals [J. Reine Angew. Math. 313, 161--170 (1980; Zbl 0412.10030)] as used in the earlier work of the author with \textit{K. Matsumoto, Y. Tanigawa} [Math. Proc. Camb. Philos. Soc. 127, 117--131 (1999; Zbl 0958.11065)]. Further the author studies the corresponding Riesz mean of order 1, \(\Delta_1(x)=\int_0^x\Delta(u)\,du\), and obtains a bound for the fourth moment \[ \int_0^X\Delta_1^4(x)\,dx \ll_\varepsilon X^{{11\over 2}+\varepsilon} \] using a recent Diophantine inequality in four variables obtained by \textit{O. Robert} and \textit{P. Sargos} [J. Reine Angew. Math. 591, 1--20 (2006; Zbl 1165.11067)] in their study of exponential sums. That this bound is best possible up to \(\varepsilon\) follows from the Ivić, Matsumoto, Tanigawa asymptotic formula for the mean square of \(\Delta_1(x)\) with an error term \(\mathcal{O}(x^{3+\varepsilon})\). Reviewer's remark: There is a misprint in reference [14]: the author is P. Shiu and the title \textit{A Brun-Titchmarsh theorem...} instead of P. Shiu and A. Brun and the title \textit{Titchmarsh theorem...'}!