A nonvanishing result for twists of L-functions of GL(n)
© 1994 Duke University Press. Received 6 October 1993. Revision received 30 November 1993. We thank the following people: D. Rohrlich for his encouragement and for reading the paper carefully, which led to his discovering an error in the first version, H. Jacquet for very useful comments and conversations on L-functions, F. Shahidi and V. K. Murty for helpful remarks, S. Friedberg for suggestions for improvement, and P. Sarnak for raising the problem of seeing how much better one could do by assuming the Ramanujan conjecture. As it is evident from the statement of the Theorem above, it helps some, but not too much, to assume this conjecture for π, which says that the Hecke eigenvalues a(n, π)^2 are bounded. (Remember the unitary normalization.) The Rankin method shows that the average value of la(n, π)^2 is O(1), and this is nearly as good. Finally, the first author would like to take this opportunity to acknowledge the hospitality of the California Institute of Technology mathematics department during her stay there, when the bulk of this work was completed. The second author would like to thank the Hebrew University for an invitation to visit for two weeks during which time this paper was essentially finalized.