In this post we consider the Ramanujan tau function $\tau(n)$, see the Wikipedia *Ramanujan tau function*, and we consider partial sums of its corresponding Dirichlet series (see for example the article *Tau Dirichlet Series* from the encyclopedia Wolfram MathWorld)

$$\varphi_N(s):=\sum_{n=1}^N\frac{\tau(n)}{n^s},\tag{1}$$ where thus $N>1$ is a positive integer, and $s=x+iy$ denotes the complex variable.

Question 1.How do you determine and calculate an approximation of a zero of $$\varphi_3(s)=\sum_{n=1}^3\frac{\tau(n)}{n^s}?$$Many thanks.

I'm inspired in section 4 from [1], thus I know that at least one can to deduce these equations $$0=1-24\cdot 2^{-x}\cos(y \log 2)+252\cdot 3^{-x}\cos(y\log 3)$$ and

$$0=-24\cdot 2^{-x}\sin(y \log 2)+252\cdot 3^{-x}\sin(y\log 3),$$

but I don't know how get an approximation for one of those zeros $s=x+iy$ or if this is a good way.

Question 2.Inspired in section 3 I would like to know if it is possible to state or conjecture a region for which the partial sum of the Ramanujan's zeta function $\varphi_N(s)$, for $N$ large enough, doesn't vanish (see the quoted statement due to Montgomery in page 23).Many thanks.

If some of previous questions are in the literature feel free to refer it, and I try to search and read the answers to my questions from the literature. I hope that both questions have mathematical sense.

## References:

[1] Peter Borwein, Greg Fee, Ron Ferguson and Alexa van der Waall, *Zeros of Partial Sums of the Riemann Zeta Function*, Experimental Mathematics, Vol. 16 (2007), No. 1, A K Peters, Ltd.