# Langlands' original observation about Ramanujan conjecture

Obviously functoriality of arbitrary high symmetric power lifts of automorphic forms on GL(2) will lead to the Ramanujan conjecture. But I guess that is too strong for Ramanujan. I came across some statement online some months back that Langlands observed something much weaker(but still about symmetric powers) will also imply Ramanujan conjecture.

What's Langlands' original observation?

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In Section 5.2 of The role of the Ramanujan Conjecture in analytic number theory by V. Blomer and F. Brumley, Bulletin AMS 50 (2013) 267--320, the authors write:

There is perhaps no better illustration of the fundamental role of $L$-functions in this subject than the observation (due to Langlands) that the absolute convergence of $L(s,\pi,\mathrm{sym}^k)$ on $\mathrm{Re}(s)>1$ for all $k\ge 2$ implies the Ramanujan conjecture for $\mathrm{GL}_2$.

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One really needs only the holomorphy of these symmetric powers in Re$(s)>1$ (rather than absolute convergence -- of course, once Ramanujan follows then holomorphy becomes absolute convergence). The letter from Serre to Deshouillers which is reproduced in the Blomer-Brumley article gives a sketch argument, which Serre attributes to Langlands and Deligne. See also page 666 of this article by Sarnak: web.math.princeton.edu/sarnak/FieldNotesCurrent.pdf –  Lucia Jun 4 '14 at 22:21
Thanks for the correction. I am not an expert at all, so I just copied the statement. –  Aurel Jun 4 '14 at 22:56

Aurel has already given a good answer to this, and as I note in my comment to his answer one needs only the holomorphy of $L(s,\pi, \text{sym}^k)$ in the region Re$(s)>1$ to obtain the Ramanujan conjecture. The argument is described in Section 8 of Langlands's paper: Problems in the theory of automorphic forms.

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