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

Full-text PDF Free Access

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.

**[1]**P. Deligne,*La conjecture de Weil I*, Publ. Math. I.H.E.S.**43**, 273--307 (1974). MR**49:5013****[2]**J. -M. Deshouillers and H. Iwaniec,*Kloosterman sums and Fourier coefficients of cusp forms*, Invent. Math.**70**, 219--288 (1982). MR**84m:10015****[3]**J.-M. Deshouillers and H. Iwaniec,*The non-vanishing of the Rankin-Selberg zeta-functions at special points*, in ``The Selberg Trace Formula and Related Topics,'' Contemp. Math. 53, Amer. Math. Soc., Providence, R. I., 51--95 (1986). MR**88d:11047****[4]**T. Estermann,*On Kloosterman's sums*, Mathematika**8**, 83--86 (1961). MR**23:A3716****[5]**W. Luo,*On the non-vanishing of Rankin-Selberg L-functions*, Duke Math. J.**69**, 411--425 (1993). MR**93m:11040****[6]**R. Phillips and P. Sarnak,*On cusp form for cofinite subgroups of*, Invent. Math.**80**, 339--364 (1985). MR**86m:11037****[7]**R. A. Rankin,*Contributions of the theory of Ramanujan's functions and similar arithmetical functions*, Proc. Cambridge Phil. Soc.**35**, 357--372 (1939). MR**1:69d****[8]**G. Shimura,*On the holomorphy of certain Dirichlet series*, Proc. London Math. Soc.**31**, 79--98 (1975). MR**52:3064****[9]**G. Shimura,*Introduction to the Arithmetic Theory of Automorphic Functions*, Princeton Univ. Press, 1971. MR**47:3318****[10]**E. C. Titchmarsh,*The Theory of the Riemann Zeta-Function*, Oxford, 1950. MR**13:741c****[11]**A. Weil,*On some exponential sums*, Proc. Nat. Acad. Sci.**34**, 204--207 (1948). MR**10:234e**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (1991):
11M26,
11F11

Retrieve articles in all journals with MSC (1991): 11M26, 11F11

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