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

DOI:
https://doi.org/10.1090/S0002-9947-96-01540-1

MathSciNet review:
1333393

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Abstract | References | Similar Articles | Additional Information

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 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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Additional Information

**Xian-jin Li**

Affiliation:
Department of Mathematics, Purdue University, West Lafayette, Indiana 47907

Email:
xianjin@math.purdue.edu

DOI:
https://doi.org/10.1090/S0002-9947-96-01540-1

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