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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On the Poles of Rankin-Selberg Convolutions of Modular Forms

Author: Xian-jin Li
Journal: Trans. Amer. Math. Soc. 348 (1996), 1213-1234
MSC (1991): Primary 11M26, 11F11
MathSciNet review: 1333393
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Abstract: The Rankin-Selberg convolution is usually normalized by the multiplication of a zeta factor. One naturally expects that the non-normalized convolution will have poles where the zeta factor has zeros, and that these poles will have the same order as the zeros of the zeta factor. However, this will only happen if the normalized convolution does not vanish at the zeros of the zeta factor. In this paper, we prove that given any point inside the critical strip, which is not equal to $\frac {1}{2}$ and is not a zero of the Riemann zeta function, there exist infinitely many cusp forms whose normalized convolutions do not vanish at that point.

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Xian-jin Li
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907

Keywords: Poles, Rankin-Selberg convolutions, zeros, Riemann zeta function, cusp forms
Received by editor(s): January 17, 1994
Received by editor(s) in revised form: April 13, 1995
Article copyright: © Copyright 1996 American Mathematical Society