I'm reading Soundararajan's <https://arxiv.org/pdf/0705.0723.pdf>, and on page 5, one has


$$\sum\_{n\leq x} \frac{\Lambda(n)}{n^z} \log (x/n) = -\frac{\zeta'}{\zeta}(z)\log x - \Big(\frac{\zeta'}{\zeta}(z) \Big)' -\sum\_{\rho} \frac{x^{\rho-z}}{(\rho-z)^2} + O(1/T),$$
where $\Lambda$ denotes the von Mangoldt function, $\zeta$ the Riemann zeta function, $\zeta(\rho)=0, \Re(z) \in (1/2, 2], x\geq 2$ and $|\Im(\rho)|\leq T$. Maybe i'm missing something, but doesn't the right hand side of the above formula have a pole at $z=1$, which would render the formula meaningless at $z=1$?



On Soundararajan's explicit formula
The formula you quote is on page 8 (not page 5). The sum is over all zeros $\rho$, not just those with $|\Im(\rho)|\leq T$. On the other hand, there is the assumption $\Im(z)\in[T,2T]$.


