Fourth moments of \(L\)-functions attached to newforms. (English) Zbl 0659.10025
Let \(\mathcal N_k\) denote the set of normalized newforms of weight \(k\) on \(\Gamma_0(2)\) and \(L_f(s)\) the usual Dirichlet series attached to \(f\in\mathcal N_k\). The author derives an estimate for the fourth moments of \(L\)-functions, namely
\[
\sum_{\substack{8\leq k\leq T,\\ k\text{ even}}} \sum_{f\in\mathcal N_k} \int^{T}_{-T} \vert L_f(k+it)\vert^4 \,dt \ll T^3 \log^4T\quad\text{as } T\to \infty.
\]
Therefore he demonstrates a result on the mean square of Dirichlet series attached to Maass wave forms on the \((n+1)\)-dimensional hyperbolic space, which were investigated by H. Maaß [Abh. Math. Semin. Univ. Hamb. 16, No. 3/4, 72–100 (1949; Zbl 0034.34801)]. The case \(n=2\) is basically due to P. Sarnak [Commun. Pure Appl. Math. 38, 167–178 (1985; Zbl 0577.10026)]. Then the case \(n=4\) is applied. The main result follows from a tedious calculation of the average of the divisor function on the Hurwitz order of integral quaternions.
Reviewer: Aloys Krieg (Aachen)
MSC:
| 11F66 | Langlands \(L\)-functions; one variable Dirichlet series and functional equations |
| 11R52 | Quaternion and other division algebras: arithmetic, zeta functions |
Keywords:
newforms; Dirichlet series; fourth moments of L-functions; mean square; Maass wave forms; hyperbolic space; divisor function; Hurwitz order of integral quaternionsReferences:
| [1] | Möbius transformations and Clifford numbers, in Differential Geometry and Complex Analysis, Vol. dedic. H. E. Rauch, Springer-Verlag, New York, 1985. pp. 63–73. |
| [2] | Atkin, Math. Ann. 185 pp 134– (1986) |
| [3] | and , Asymptotic Expansions of Integrals, Dover, New York, 1986. |
| [4] | Chandrasekharan, Math. Ann. 152 pp 30– (1963) |
| [5] | Deshouillers, Inv. Math. 70 pp 219– (1982) |
| [6] | Some problems in multi-dimensional analytic number theory, to appear in Acta Arithmetica. |
| [7] | The Basis Problem for Modular Forms and the Traces of the Hecke Operators, Lecture Notes in Mathematics 320, Springer-Verlag, Berlin, 1972, pp. 75–151. |
| [8] | Good, Mathematika 29 pp 278– (1982) |
| [9] | and , Table of Integrals and Products, Academic Press, New York, 1980. |
| [10] | Collected Papers II, Clarendon Press, Oxford, 1967. |
| [11] | Mathematische Werke, Vandenhoeck and Ruprecht, Göttingen, 1983,. |
| [12] | Lectures on Dirichlet Series, Modular Functions, and Quadratic Forms, Vandenhoeck and Ruprecht, Göttingen, 1983,. |
| [13] | Groups and Geometric Analysis, Academic Press, Orlando, 1984. |
| [14] | Vorlesungen über die Zahlentheorie der Quaternionen, Springer-Verlag, Berlin, 1919. · JFM 47.0106.01 · doi:10.1007/978-3-642-47536-8 |
| [15] | Ingham, Proc. London Math. Soc. 27 pp 273– (1926) |
| [16] | Jones, J. Math. and Physics 37 pp 1– (1958) · doi:10.1002/sapm19583711 |
| [17] | Modular Forms on Half-Spaces of Quaternions, Lecture Notes in Mathematics 1143, Springer Verlag, Berlin, 1985. · Zbl 0564.10032 |
| [18] | Special Functions and their Applications, trans. R. Silverman, Dover, New York, 1972. |
| [19] | Ergodic Properties of Algebraic Fields, Springer-Verlag, New York, 1968. · doi:10.1007/978-3-642-86631-9 |
| [20] | Maass, Abh. Math. Seminar Hansischen Univ. 16 pp 72– (1949) · Zbl 0034.34801 · doi:10.1007/BF03343519 |
| [21] | Pizer, J. Math. Soc. Japan 28 pp 689– (1976) |
| [22] | Pizer, J. of Algebra 64 pp 340– (1980) |
| [23] | Potter, Proc. Lon. Math. Soc. 47 pp 1– (1945) |
| [24] | Rankin, Duke Math. J. 12 pp 281– (1945) |
| [25] | Modular Forms and Functions, Cambridge University Press, Cambridge, 1977. · Zbl 0376.10020 · doi:10.1017/CBO9780511566035 |
| [26] | Sarnak, Comm. Pure Appl. Math. 38 pp 167– (1985) |
| [27] | Lectures on Advanced Analytic Number Theory, Tata Inst. Bombay, 1961. |
| [28] | The Theory of the Riemann Zeta Function, Clarendon Press, Oxford, 1951. |
| [29] | Vahlen, Math. Ann. 55 pp 585– (1902) |
| [30] | Special Functions and the Theory of Group Representations, Trans, of MaTh. Monographs 22, AMS, Providence, 1968,. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
