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:
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[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).