Why is $\Re \zeta$ non-zero on the line $\Re(s)=\frac32$

Question

I am trying to understand why $\Re \zeta (s)$ is non-zero on the line $\Re (s)=\frac32$. This in stated without explanation in section 6.6 of the book ‘Riemann’s Zeta Function’ by H.M. Edwards, and it is a key part in Backlund’s proof for the evaluation of $N(T)$ (number of complex zeta roots below $t=T$).

Using the zeta formula, I have deduced for $s=3/2 +iT$, we have

$$ \Re \zeta (s)= \sum_{n=1}^{\infty} \frac{\cos(T \log n)}{n^{3/2}}.$$

How can one show this summation must be non-zero?

Answer 1

Here's the proof from Edwards' book, p129: