bounding gaps between points in an interval

Question

I've been reading Davenport's Multiplicative Number Theory and came across something that I didn't understand. On p. 108, there is an argument for finding a lower bound on the imaginary parts $\gamma$ of roots of $\zeta(s)$ that have $|\gamma - T| < 1$ for some $T > 0$. A result in a previous chapter states that the number of zeros with $|\gamma -T| < 1$ is $O(\log T)$. This implies that there is some pair of roots with $|\gamma - T| < 1$ that are distance $\Omega(\log T)^{-1}$ apart. Then, it is stated that varying $T$ by some $``\text{bounded}"$ amount ensures that $|\gamma - T|$ is $\Omega(\log T)^{-1}$ for all zeros $\beta + i\gamma$ of $\zeta(s)$ such that $|\gamma - T| < 1$. How does the last statement follow from the previous arguments? Also, what is meant by a bounded amount?

Answer 1

For sufficiently large $T$, the number of zeros in the interval $|\gamma - T| < 1$ is less than $c \log T$ for some positive constant $c$.

Among the zeros that are not in the interval $|\gamma - T| < 1$, it is immediate that, for those zeros, $$|\gamma - T| \geq 1 \gg (\log T)^{-1}.$$

Now consider the zeros that are in the interval $|\gamma - T| < 1$. The number of these is less than $c \log T$. Therefore among some pair of consecutive imaginary parts of the zeros, say $\gamma_{n-1}$ and $\gamma_n$, we must have a gap of greater than $ (c \log T)^{-1}$.

What about the values of $|\gamma - T|$ for zeros in the interval $|\gamma - T| < 1$? Well it is certainly possible that our choice of $T$ could have been unlucky, and we landed exactly on a zero. But if we change our $T$ by a bounded amount, i.e. less than $1/2$, we can put $T$ halfway between $\gamma_{n-1}$ and $\gamma_n$, ensuring that $$|\gamma -T| > (2c \log T)^{-1}$$ for all zeros $\beta + i\gamma$.