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 $f=\overline{g}$?
  • 1
    $\begingroup$ I'd say : add finitely many Euler factors to $L(f\otimes g, s)$ to obtain a Dirichlet series $H(s)$ with functional equation. Obtain a bound $H(1+it) = O(\log^m(t))$ (in the same way you showed the PNT for $L(f,s)$ ?). Use the functional equation to obtain a bound for $H(it)$. Apply the convexity theorem. I don't know if the implied constants are effective. What the found for $L(f\otimes g,\frac12+it)$ would be useful for ? $\endgroup$ – reuns Nov 11 '18 at 21:55
  • $\begingroup$ Thanks for your comment. I use the bound as a factor for a more general result which applies not only to newforms but to arbitrary cusp forms. The readers can insert a stronger bound if they want quantitatively better results but I just need an elementarily introduced placeholder instead of a reference to an article of 30 pages about any subconvexity bound. $\endgroup$ – Nodt Greenish Nov 12 '18 at 10:21
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    $\begingroup$ By the way, $L(s,f \otimes g)$ is not equal to $\sum_{n = 1}^{\infty} \frac{\lambda_f(n) \lambda_g(n)}{n^s}$. Rather, it is equal (up to some Euler factors at bad primes) to this Dirichlet series times $L(2s,\chi_f \chi_g)$, where $\chi_f, \chi_g$ are the nebentypen of $f$ and $g$. $\endgroup$ – Peter Humphries Nov 12 '18 at 13:06
  • $\begingroup$ Alright, so I need a factor of $\zeta(2s)$. Thanks for that reminder @PeterHumphries, I will watch out for that in my notation. $\endgroup$ – Nodt Greenish Nov 12 '18 at 13:52
  • $\begingroup$ @reuns I define $H(s)=(2\pi)^{-2s}\Gamma(s+k-1)\Gamma(s)L(s,f\otimes g)$ which allows the functional equation $H(1-s)=H(s)$ for $f\neq g$. So for $L(f\otimes f,s)$ I need another estimation... $\endgroup$ – Nodt Greenish Nov 12 '18 at 15:17

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.


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