I am currently reading Riemann Zeta Function by Katsuraba and Voronin
Theorem : Let $\rho_n = \beta_n + \gamma_n,$ where $1,2,\dots$ be all complex zeros of $\zeta(s)$, and let $T \ge 2$. Then $$\sum_{n=1}^{\infty} \frac{1}{1+(T-\gamma_n)} \leq c \log T.$$
Question: Can someone explain why the author used $\frac{1}{2}$, here instead of $1$? ($\text{Re} \frac{1}{s-\rho_n} \ge \frac{0.5}{1+(T-\gamma_n)^2}$.) Is there any intuition why $1/2$ will work better here? Or is this just a typo?.