Convexity Bound of Rankin-Selberg L-Function

by Nodt Greenish   Last Updated November 13, 2018 18:20 PM

Let $f,g$ be primitive modularforms of arbitrary levels $N_1,N_2$, trivial nebentypus and same weight $k$. Let $L(f\otimes g,s)=\zeta(2s)\sum_{n\geq1}\frac{\lambda_f(n)\lambda_g(n)}{n^s}$ be the Rankin-Selberg L-Function where $\lambda_f$ and $\lambda_g$ are the respective normalized Hecke-Eigenvalues.

I'm looking for a simple upper bound of $L(f\otimes g,\frac12+it)$ in the level aspect which holds under general assumptions as given above. In nearly every piece of literature on subconvexity bounds (e.g. Introduction of "Rankin Selberg L-functions in the level aspect" by Kowalski, Michel, Vanderkam) there is the hint that the convexity bound ($\sqrt N$ in the level aspect, $k$ in the weight aspect) is achieved by applying the Phragmén-Lindelöf principle. As far as I understood, this requires another bound of the function on the edges of the vertical strip $0<\Re(s)<1$. So my questions are

  • Is there an elementary proof to this problem?
  • What bound can I use for $L(f\otimes g, 1)$?
  • How do I take care of the residue in $s=1$ for $f=\overline{g}$?

Answers 1

To provide a brief answer which can be found in "Analytic Number Theory" of Iwaniec and Kowalsky:

We use the absolute convergence of $L(f,s)$ in $\sigma>1$ (or the bound $|\lambda(n)|\leq \tau(n)$), the functional equation, Stirling's estimate for the gamma function and the Phragmen-Lindelöf convexity principle.

All mentioned formulas can be found in the book as well.

Nodt Greenish
Nodt Greenish
November 13, 2018 18:17 PM

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