# absolute convergence of Rankin-Selberg series

Let $\pi$ and $\pi'$ be two general automorphic representation on $GL(n)$ and $GL(n')$ over $\mathbb{Q}$.

I heard that the rankin-selberg convolution L-function $L(s,\pi\times\pi')$ is absolutely convergent in {$Re(s)>1$}. I can only convince myself for the special case that $\pi$ and $\pi'$ are contragredient.

Why is this true for other cases?

• For what range of $s$? The Dirichlet series is clearly divergent for $\operatorname{Re}(s) \ll 0$. – David Loeffler Jun 6 '14 at 20:23
• Essentially Cauchy-Schwarz allows you to get the absolute convergence for $\pi \times \pi^{\prime}$ from knowing $\pi \times \overline{\pi}$ and ${\pi^{\prime}} \times \overline{\pi^{\prime}}$. See Molteni's paper in Duke Math. J. – Lucia Jun 6 '14 at 22:46
• @Lucia: My response below is an elaboration of your comment. Thanks! – GH from MO Jun 7 '14 at 13:08
• In fact absolute convergence was already proved by Jacquet-Shalika in general: see (5.3) Theorem on page 555 in Amer. J. of Math. 103 (1981). – GH from MO Jun 9 '14 at 6:39

This is an elaboration of Lucia's comment. Let us consider the Dirichlet coefficients of $L(s,\pi\times\pi')$, $L(s,\pi\times\tilde\pi)$, $L(s,\pi'\times\tilde\pi')$ at a prime power $p^k$. Following the proof of Proposition 6 in Molteni's paper (Duke Math. J. 111 (2002), 133-158), we see that there exists a polynomial $P_k\in\mathbb{N}[x_1,\dots,x_k]$ such that $$k!\ a_{\pi\times\pi'}(p^k) = P_k(\sigma_1\tau_1,\dots,\sigma_k\tau_k),$$ $$k!\ a_{\pi\times\tilde\pi}(p^k) = P_k(|\sigma_1|^2,\dots,|\sigma_k|^2),$$ $$k!\ a_{\pi'\times\tilde\pi'}(p^k) = P_k(|\tau_1|^2,\dots,|\tau_k|^2),$$ where $\sigma_h$ (resp. $\tau_h$) denotes the $h$-th power sum of the Langlands parameters of $\pi$ (resp. $\pi'$) at $p$. Using Cauchy-Schwarz, it follows that $$|a_{\pi\times\pi'}(p^k)|\leq \sqrt{a_{\pi\times\tilde\pi}(p^k)a_{\pi'\times\tilde\pi'}(p^k)}.$$ Note that the coefficients on the right are nonnegative. Hence for all positive integers $n$, we have $$|a_{\pi\times\pi'}(n)|\leq \sqrt{a_{\pi\times\tilde\pi}(n)a_{\pi'\times\tilde\pi'}(n)}\leq (a_{\pi\times\tilde\pi}(n)+a_{\pi'\times\tilde\pi'}(n))/2.$$ It follows that the absolute Dirichlet series for $L(s,\pi\times\pi')$ is majorized by the average of the Dirichlet series for $L(\sigma,\pi\times\tilde\pi)$ and $L(\sigma,\pi'\times\tilde\pi')$. The latter converges for $\sigma>1$ by Landau's lemma, hence $L(s,\pi\times\pi')$ converges absolutely for $\Re(s)=\sigma>1$.
Added 1. The above argument requires that $p$ is an unramified prime for $\pi$ and $\pi'$, hence also that $n$ is only divisible by such primes. This does not alter the final conclusion though, because omitting or including finitely many Euler factors does not alter absolute convergence of the corresponding Dirichlet series in the half-plane $\Re(s)>1$.
Added 2. For a related and more complete result concerning the logarithmic derivative of Rankin-Selberg $L$-functions, see Brumley's Appendix to the exciting new preprint of Soundararajan and Thorner.
• A bit off-topic, but if I'm not mistaken, I was told that the non vanishing on the $\sigma=1$ line of these L functions was needed to ensure unique factorization. Has it been proved that these L functions don't vanish on the considered line? I can ask this question in a new thread if needed. – Sylvain JULIEN Jun 7 '14 at 19:39
• @Sylvian: Automorphic forms factor into local representations over the primes, which yields the Euler product factorization of $L(s,\pi)$ and $L(s,\pi\times\pi')$. This has nothing to do with non-vanishing on $\sigma=1$. Regarding non-vanishing, it was proved by Jacquet--Shalika in 1976 that $L(s,\pi)\neq 0$ on $\sigma=1$. I think the same is not known for $L(s,\pi\times\pi')$, but of course by the Langlands conjectures these $L$-functions are also of the form $L(s,\Pi)$ with a single $\Pi$, so non-vanishing should hold for them as well. – GH from MO Jun 7 '14 at 20:12
• @Sylvain: Factoring an automorphic $L$-function or a Rankin-Selberg $L$-function into primitive elements of the Selberg class is off-topic here, especially that these $L$-functions themselves are not known to belong to the Selberg class. – GH from MO Jun 7 '14 at 20:58
• @SylvainJULIEN: I told you a while ago that $L(s,\pi\times\pi')\neq 0$ on $\sigma=1$ is not known. I was wrong, Shahidi proved this nonvanishing, see Theorem 5.2 on page 353 in Amer. J. of Math. 103 (1981). – GH from MO Nov 2 '16 at 7:30