×

Zeros of L-functions attached to Maass forms. (English) Zbl 0565.10026

Considered are the L-functions corresponding to even real analytic cusp forms of weight zero for the modular group, eigenfunctions of all Hecke operators. It is shown that the number of zeros s of such an L-function on the critical line, satisfying \(| Im s| \leq T\), is \(\ll T.\)
In the proof of this result an explicit formula is used, reconstructing the cusp form from the L-function everywhere on the upper half plane. In the well known Maass theory the inverse Mellin transform reconstructs the cusp form on \(\{\) iy\(| y>0\}\) only; so this new representation is of independent interest.
For the Fourier coefficients the estimate \(\sum_{1\leq n\leq T}a_ n e^{2\pi inx}\ll_{\epsilon}T^{+\epsilon}\) is given. I distrust its proof, as an improbable asymptotic expansion of \(K_{ir}(t)^{-1}\) is used. - Minor corrections: In (4.5) insert a factor \(\ell +1\). In (5.11) take \(\int^{q}_{0}+\int^{1}_{1-q}\) with \(q=(\log n)^{\alpha}/(n+1)\) instead of \(\int^{q}_{0}\), and change (5.12) accordingly.
Reviewer: R.W.Bruggeman

MSC:

11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11F11 Holomorphic modular forms of integral weight
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] Chandrasekharan, K., Narasimhan, R.: Zeta-functions of ideal classes in quadratic fields and zeros on the critical line. Comment. Math. Helv.43, 18-30 (1968) · Zbl 0157.09302 · doi:10.1007/BF02564377
[2] Cohen, P., Sarnak, P.: Discrete groups and geometry. (to appear)
[3] Good, A.: Beiträge zur Theorie der Dirichletreihen, die Spitzenformen zugeordnet sind. J. Number Theory13, 18-65 (1981) · Zbl 0446.10022 · doi:10.1016/0022-314X(81)90028-7
[4] Gradshteyn, I., Ryzhik, I.: Tables of integrals and products. New York-London: Academic Press 1980 · Zbl 0521.33001
[5] Hafner, J.L.: Zeros on the critical line for Dirichlet series attached to certain cusp forms. Math. Ann.264, 21-37 (1983) · Zbl 0507.10024 · doi:10.1007/BF01458048
[6] Hardy, G.H., Littlewood, J.E.: The zeros of Riemann’s zeta function on the critical line. Math. Zeit.10, 283-317 (1921) · JFM 48.0344.03 · doi:10.1007/BF01211614
[7] Hejhal, D.A.: The Selberg trace formula forPSL(2,?) Vol.2, Lecture Notes in Mathematics1001. Berlin-Heidelberg-New York-Tokyo: Springer 1983
[8] Jacquet, H., Shalika, J.: A non-vanishing theorem for zeta functions onGL n . Invent. Math.38, 1-16 (1976/77) · Zbl 0349.12006 · doi:10.1007/BF01390166
[9] Kubota, T.: Elementary theory of Eisenstein series. Tokyo: Kodansha 1973 · Zbl 0268.10012
[10] Lekkerkerker, C.G.: On the zeros of a class of Dirichlet series. Dissertation, Utrecht, 1955 · Zbl 1173.30301
[11] Maass, H.: Über eine neue Art von nicht analytischen automorphen Funktionen und die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen. Math. Ann.121, 141-183 (1949) · Zbl 0034.31702 · doi:10.1007/BF01329622
[12] Moreno, C.J.: Prime number theorems for the coefficients of modular forms. Bull. Amer. Math. Soc.78, 796-798 (1972) · Zbl 0274.10031 · doi:10.1090/S0002-9904-1972-13040-4
[13] Rankin, R.A.: Contributions to the theory of Ramanujan’s function ?(n) and similar arithmetical functions I, II. Proc. Cambridge Phil. Soc.35, 351-372 (1939) · JFM 65.0353.01 · doi:10.1017/S0305004100021095
[14] Sarnak, P.: Fourth moments of Grossencharakter zeta functions (to appear)
[15] Selberg, A.: On the zeros of Riemann’s zeta-function. Skr. Nor. Vidensk. Akad. Oslo I,10, 59, (1942) · JFM 68.0161.01
[16] Stark, H.: Fourier coefficients of Maass wave forms (to appear)
[17] Titchmarsh, E.C.: The theory of the Riemann zeta function. London, New York: Oxford University Press (Clarendon) 1951 · Zbl 0042.07901
[18] Venkov, A.B.: On an asymptotic formula connected with the number of eigenvalues corresponding to odd eigenfunctions of the Laplace-Beltrami operator on a fundamental region of the modular groupPSL(2,?), Dokl. Akad. Nauk SSSR233 1021-1023 (1977); English transl. in Soviet Math. Dokl.18 (1977)
[19] Venkov, A.B.: Spectral theory of automorphic functions. Proc. Steklov Inst. Math. (1982) Russian Tom153 (1981) · Zbl 0483.10029
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.