Some remarks on odd Maass wave forms (and a correction to [1]). (English) Zbl 0628.10033
This note extends to odd Maass forms the results in C. Epstein, J. L. Hafner and P. Sarnak [Zeros of L-functions attached to Maass forms; Math. Z. 190, 113-128 (1985; Zbl 0565.10026)]. These results are an integral representation giving real analytic modular forms in terms of their L-function everywhere on the upper half plane, and the fact that the number of zeros s on the critical line of the L-function with \(| Im s| \leq T\) is \(\gg T\). Further the estimate \(\sum_{| n| \leq T}a_ n e^{2\pi inx} \ll_{\epsilon} T^{+\epsilon}\) in theorem 3.1 of loc. cit. is proved.
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 |
Keywords:
inverse Mellin transform; odd Maass forms; real analytic modular forms; L-function; number of zeros; critical lineCitations:
Zbl 0565.10026References:
| [1] | Epstein, C., Hafner, J.L., Sarnak, P.: Zeros ofL-functions attached to Maass forms. Math. Z.190, 113-128 )1985) · Zbl 0565.10026 · doi:10.1007/BF01159169 |
| [2] | Gradshteyn, I., Ryzhik, I.: Tables of integrals and products. New York-London: Academic Press 1980 · Zbl 0521.33001 |
| [3] | Hafner, J.L.: On the zeros of Maass wave formL-functions. Bull. A.M.S. (N.S.)15, 61-64 (1986) · Zbl 0603.10027 · doi:10.1090/S0273-0979-1986-15434-0 |
| [4] | 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 |
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