Average bounds on Rankin-Selberg coefficients for modular forms

Question

Let $f$ be a cuspidal Hecke newform of weight $k$ and level $N$, and denote by $a_f(n)$ its $n$-th Fourier coefficient. The newform $f$ is normalized so that $a_f(1) = 1$. As a consequence of Rankin-Selberg theory, we have that $$ \sum_{p \leq x} \frac{a_f(p)^2}{p} = \log \log x + B + O(1/\log x), $$ for any large $x$, where $B$ is a constant. What do we know about the dependence of $B$ and the implicit constant in the level aspect? Are they bounded in the level aspect, or at least not growing faster than $\log \log N$? (I would like to apply it with $x$ of size about $N$)

To state an analogue result, we can prove that for a compactly supported function $\phi$ we have $$ \sum_{\substack{p \leq N}} \frac{a_f(p)^2}{p} \frac{\log(p)^2}{\log(N)^2} \phi\left( \frac{\log p}{\log N}\right) = B_\phi + O\left( \frac{\log\log N}{\log N}\right) $$ where $B_\phi$ only depends on $\phi$ but not on $f$, and the implicit constant in the error term is absolute. Is there a similar statement for the first sum above? Or a proper way to deduce the first equation from the second statement?

Answer 1

In Mertens' original result, he proves that $$\sum_{p\leq x}\frac{1}{p}= \log\log x+B+O\Big(\frac{1}{\log x}\Big),$$ where $$B = \sum_p\Big(\log\Big(1-\frac{1}{p}\Big)-\frac{1}{p}\Big) = - \sum_p\sum_{k=2}^{\infty}\frac{1}{k p^k}.$$ Note that the summands in the sum up to $x$ and the summands in the sum defining $B$ are the Dirichlet coefficients of $\log \zeta(s+1)$, which are $\Lambda(n)/(n\log n)$ for $n\geq 2$. Here, $\Lambda(n)$ is the von Mangoldt function.

It seems reasonable to suspect that if $f$ is self-dual (as the wording of the problem suggests, but does not explicitly state), then $$\sum_{p\leq x}\frac{a_f(p)^2}{p}=\log\log x+ B_f+O_f\Big(\frac{1}{\log x}\Big),$$ where $$B_f = - \sum_{\text{$n$ composite}}\frac{\Lambda_{f\times f}(n)}{n\log n}-\sum_{p\mid N}\frac{a_f(p)^2}{p}+\sum_{p\mid N}\frac{a_{f\times f}(p)}{p},$$ where $$\sum_{n=1}^{\infty}\frac{\Lambda_{f\times f}(n)}{n^s} = -\frac{L'}{L}(s,f\times f)$$ (the logarithmic derivative of the Rankin-Selberg $L$-function $L(s,f\times f)$). Note that if $p$ is a prime not dividing the level, then $\Lambda_{f\times f}(p) = a_f(p)^2\log p$. The coefficients $a_f(p)^2$ and $a_{f\times f}(p)$ might be different at primes $p$ dividing $N$.

Since such $f$ satisfy Ramanujan's conjecture, we have $|\Lambda_{f\times f}(n)|\leq 4\Lambda(n)$. So $B$ can be bounded independently of $N$ or $k$.

I have not supplied a proof, but the proof is at least as complicated as the case of $\zeta(s)$, which is detailed here. The proof will generalize.

ADDED: In analogy with the case of primes, the Dirichlet coefficients of $\log L(s+1,f\times f)$ are $\Lambda_{f\times f}(n)/(n\log n)$.