Moments of products of automorphic \(L\)-functions
Understanding the behavior of the Riemann zeta function, and more generally, zeta and \(L\)-functions possessing an Euler product on the critical line \(\mathrm{Re}(s)=1/2\) is one of fundamental problems of analytic number theory. For primitive \(L\)-functions belonging to the Selberg class and for automorphic \(L\)-functions attached to irreducible cuspidal automorphic representations of \(\mathrm{GL}_m\) over \(\mathbb{A}\) with unitary central character it is conjectured that for any \(k>0\) there exist a constant \(C(k,L)\) such that \[ \int_{0}^{T}\left| L(1/2 +it)\right|^{2k} \sim C(k,L) T (\log T)^{k^2}, \] as \(T \to \infty\). For degree-one functions in the Selberg class, i.e., the Riemann zeta function and Dirichlet \(L\)-functions the above conjecture is known to hold for \(k\in\{1,2\}\). For degree-two functions, the conjecture is known to hold with \(k=1\) for certain \(L\)-functions. Actually, it is expected that the values of distinct primitive \(L\)-functions on the critical line are uncorrelated. Therefore, it is expected that for any \(r\) distinct primitive \(L\)-functions \(L_1, \dots, L_r\) and any \(k_1, \dots, k_r >0\) there exist a constant \(C=C(k_1,\dots,k_r, L_1, \dots, L_r)\) such that \[ \int_{0}^{T} \left|L_1(1/2 + it) \right|^{2k_1} \times \cdots \times \left|L_r(1/2 + it) \right|^{2k_r} dt \sim C T (\log T)^{k_1^2 + \dots + k_r^2} \] In the paper under review, the authors prove, under the generalized Riemann hypothesis, upper bounds for moments of products automorphic \(L\)-functions, in support of the product conjecture. Namely, they prove that for \(L\)-functions \(L(s,\pi_1), \dots, L(s,\pi_r)\) attached to distinct irreducible, unitary cuspidal automorphic representations \(\pi_1, \dots, \pi_r\) of \(\mathrm{GL}_{m_1}(\mathbb{Q}), \dots, \mathrm{GL}_{m_r}(\mathbb{Q})\) respectively, one has \[ \int_{0}^{T} \left|L(1/2 + it, \pi_1) \right|^{2k_1} \times \cdots \times \left|L(1/2 + it, \pi_r) \right|^{2k_r} dt \ll T (\log T)^{k_1^2 + \dots + k_r^2 + \varepsilon} \] for any \(k_1, \dots, k_r >0\) and every \(\varepsilon >0\), for sufficiently large \(T\), under the assumption that for \(j \in\{1, \dots, r\}\) one has either \(m_j \leq 4\) or the hypothesis H of \textit{Z. Rudnick} and \textit{P. Sarnak} [Duke Math. J. 81, No. 2, 269--322 (1996; Zbl 0866.11050)] holds. Here the implied constant depends on \(\pi_1, \dots, \pi_r\), \(k_1, \dots, k_r\) and \(\varepsilon\). Moreover, for a finite Galois extension \(K\) of \(\mathbb{Q}\), under the generalized Riemann hypothesis for the Dedekind zeta function \(\zeta_K(s)\), the authors prove the upper bound \[ \int_{0}^{T} \left|\zeta_K(1/2 + it) \right|^{2k} dt \ll T (\log T)^{[K:\mathbb{Q}]k^2 + \varepsilon} \] for any \(k, \varepsilon>0\), for sufficiently large \(T\), where the implied constant depends upon \(K\), \(k\) and \(\varepsilon\). The authors also prove the upper bound for the variance of the coefficients \(r_K(n)\) of the Dirichlet series representation of \(\zeta_K(s)=\sum_n \frac{r_K(n)}{n^s}\) in short intervals, and upper bounds for moments of products of central values of automorphic \(L\)-functions twisted by quadratic Dirichlet characters and averaged over fundamental discriminants.
scientific article; zbMATH DE number 6308467
scientific article