| DATE: | Thursday, October 18, 2001 |
| TIME: | 4:30-5:30 PM |
| PLACE: | LN 2205 |
| SPEAKER: | Marvin Knopp (Temple University) |
| TITLE: | On the signs of the coefficients of modular forms |
In a response to a question raised by Geoffrey Mason, we prove that, under suitable (weak) conditions on a discrete group Gamma acting on the upper half-plane, the sequence {a(n)} of Fourier coefficients of a cusp form f on Gamma has infinitely many changes of sign, if the a(n) are real. If the a(n) are not real, the same conclusion holds for the real and imaginary parts of the coefficients. The proof employs (i) the Mellin transform of f; (ii) a classic (1905) result of Landau on the coefficients of Dirichlet series; (iii) the fact that sum |a(n)| diverges. The converse of this result fails: there are entire forms which are not cusp forms, with {a(n)} aving infinitely many sign changes. Thus it is of interest to characterize those entire forms having a coefficient sequence with infinitely many changes of sign. This is joint work with W. Kohnen and W. Pribitkin.