Here $s=\sigma+it$ denotes the complex variable. We denote the Ramanujan's zeta function
$$\varphi(s)=\sum\_{n=1}^\infty\frac{\tau(n)}{n^s}$$


for $\Re(s)>7$, where $\tau(n)$ is the Ramanujan tau function.


While I was studying a video by Harper, a video from YouTube with title *The Riemann zeta function in short intervals - Harper - Bourbaki - 30/03/19* from the official channel **Institut Henri Poincaré**, I wondered if a similar principle than the professor shows as *Principle I* (see also PRINCIPLE 1.3 from [2]) works in some suitable sense for the Ramanujan's zeta function $\varphi(s)$.


**Principle.** *For any* $s$ *with* $\Re(s)\geq \frac{11}{6}$ *(or at least* $\Re(s)> \frac{11}{6}$*) and* $|\Im(s)|\geq 1$ *we have* $$\varphi(s)\cdot\prod\_{\substack{p\text{ prime }\\p\leq X(s)}}\left(1-\frac{\tau(p)}{p^s}+\frac{p^{11}}{p^{2s}}\right)\simeq 1\tag{1}$$
*in some "suitable sense".*



> 
> **Question.** Does previous **Principle** involving the identity $(1)$ work for any suitable sense? Explain your words. **Many thanks.**
> 
> 
> 


Thus I am asking about the explanation of the meaning of previous **Principle** involving the identity $(1)$. I just wrote it as a similar statement than the *Principle I* from Harper's colloquium (at few first minutes of the colloquium), but I have no knowledges to know the meaning of it.


References:
-----------


[1] G. H. Hardy, *Ramanujan: Twelve lectures on subjects suggested by his life and work*, AMS Chelsea Publishing (2002).


[2] Adam J. Harper, *The Riemann zeta function in short intervals [after Najnudel, and Arguin, Belius, Bourgade, Radziwiłł, and Soundararajan]*, Séminaire BOURBAKI, 71e année, 2018–2019, no 1159 (Mars 2019).



A principle around the Ramanujan's zeta function in short intervals
It seems, essentially, that you want to estimate the $L$-function


$L(s,\Delta)=\sum\_{n=1}^{\infty}\tau(n) n^{-s}=\prod\_p (1-\tau(p)p^{-s}+p^{11-2s})^{-1}$


as a short Euler product, in the sense that there exists a reasonably small integer $X(s)>0$ such that


$L(s,\Delta)\approx\prod\_{p\leq X(s)} (1-\tau(p)p^{-s}+p^{11-2s})^{-1}$


with a reasonably small error. This question has been well-studied for $L$-functions in various families when $s$ lies on the line $\mathrm{Re}(s)=1$, though one can place $s$ elsewhere in the critical strip. The literature focusing on $\mathrm{Re}(s)=1$ is very extensive, so I'll focus on that for now.


Rather than studying this problem for a single $L$-function, it is often a bit more interesting to study the problem for, say, the family of $L$-functions associated to weight $k$ level 1 holomorphic cuspidal newforms, and understand the average and limiting behavior as $k\to\infty$. Here is an excellent paper on this matter by Lau and Wu: <https://hal.archives-ouvertes.fr/hal-00097046/document>. One could also fix the weight, say $k=2$, and pursue similar questions as the level $N$ varies. Here is an excellent paper on this matter by Cogdell and Michel: <http://www.math.osu.edu/~cogdell.1/moments-www.ps>. Both of these papers were inspired in part by the work of Granville and Soundararajan, where they considered the same problem for Dirichlet $L$-functions associated to real characters: <https://arxiv.org/abs/math/0206031>.


ADDED: I was mixing normalizations. For the normalization in the original question, $\mathrm{Re}(s)=1$ should be changed to $\mathrm{Re}(s)=13/2$


