If $s_o$ is one of the non-trivial zeros of the Riemann zeta function with $0 <Re(s_o)<1$ , we know:

$$\eta(s_o ) = \left(1-2^{1-s_o}\right) \zeta(s_o)= \sum_{n=1}^\infty \frac {(-1)^{n-1}} {n^{s_o}}=0 \tag{1}$$

Now, can we show that the following partial sum can not be zero?

$N$: Sufficiently large integer

$$\left|\sum_{n=1}^N \frac {(-1)^{n-1}} {n^{s_o}}\right|≠0 \tag{2}$$

Or can we show that the remainder term can not be zero?

$$\left|\sum_{n=N+1}^\infty \frac {(-1)^{n-1}} {n^{s_o}}\right|≠0 \tag{3}$$


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