Kloosterman sums and a mean value for Dirichlet polynomials
The Lindelöf conjecture is equivalent to the estimate, for any \(\varepsilon > 0\), \[ \int^T_0 \bigl |\zeta (1/2 + it) \bigr |^{2k} dt \leq c (\varepsilon, k) T^{1 + \varepsilon}, \quad k = 1,2,3, \dots, \] where \(\zeta\) is the Riemann zeta-function. This estimate has only been proved for \(k = 1\) and \(k = 2\). Even the proof of the above estimate for the case \(k = 3\) (the sixth power moment) would be a major advance in the subject. To study this problem one approach is to consider for a Dirichlet polynomial \(P(s) = \sum^M_{m = 1} a_m m^{- s}\), \(a_m\) complex, the mean value \[ I(T,M) = {1 \over T} \int^T_0 \bigl |P(it) \bigr |^2 \bigl |\zeta (1/2) + it) \bigr |^4 dt, \] for \(T \geq 1\). The Lindelöf conjecture is equivalent to \[ I(T,M) \ll T^\varepsilon \sum^M_{m = 1} |a_m |^2 \tag{*} \] for all \(M \leq T\), and the sixth power moment estimate would follow from this estimate for \(M = T^{1/2}\). \textit{H. Iwaniec} [J. Lond. Math. Soc., II. Ser. 22, 39-45 (1980; Zbl 0439.10026)] was the first to consider \(I(T,M)\) and he proved (*) for \(M \leq T^{1/10}\). \textit{J.-M. Deshouillers} and \textit{H. Iwaniec} [Mathematika 29, 202-212 (1982; Zbl 0506.10032)] improved this result by showing (*) holds for \(M \leq T^{1/5}\). Further, if the Selberg eigenvalue conjecture is true their method allowed \(M \leq T^{1/4}\). The present paper refines one aspect of the work of Deshouillers and Iwaniec to show that in the most interesting cases one can obtain this last result unconditionally by proving that \[ I(T,M) \ll T^\varepsilon (1 + M^2 T^{- 1/2}) M \max_{m \leq M} |a_m |^2. \] This result implies (*) for \(M \ll T^{1/4}\) provided the \(a_n\) satisfy \(a_n \ll_\varepsilon n^\varepsilon\) and \(\sum^n_{m = 1} |a_m |\gg_\varepsilon n^{1 - \varepsilon}\). This is the case in the most interesting examples where \(a_n\) is \(n^{ix}\) or \(\Lambda (n)\) or \(\tau (n)\). The author mentions that the new result should have applications to questions involving primes. The proof uses a new weighted large sieve inequality for the exceptional eigenvalues of the non-Euclidean Laplacian for Hecke congruence subgroups of the full modular group. The bulk of the paper consists of the proof of this large sieve estimate which involves the use of the Kuznetsov summation formula and much of the recent machinery used to manipute the resulting sums of Kloosterman sums. The author in his proof corrects an inconsequential slip in the earlier paper of Deshouillers and Iwaniec.
scientific article; zbMATH DE number 792476