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Nonvanishing of L-functions, the Ramanujan Conjecture, and Families of Hecke Characters
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- Journal:
- Canadian Journal of Mathematics / Volume 65 / Issue 1 / 01 February 2013
- Published online by Cambridge University Press:
- 20 November 2018, pp. 22-51
- Print publication:
- 01 February 2013
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- Article
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We prove a nonvanishing result for families of
$\text{G}{{\text{L}}_{n}}\times \text{G}{{\text{L}}_{n}}$ Rankin–Selberg 
$L$-functions in the critical strip, as one factor runs over twists by Hecke characters. As an application, we simplify the proof, due to Luo, Rudnick, and Sarnak, of the best known bounds towards the Generalized Ramanujan Conjecture at the infinite places for cusp forms on 
$\text{G}{{\text{L}}_{n}}$. A key ingredient is the regularization of the units in residue classes by the use of an Arakelov ray class group.
