I am studying the Riemann zeta function. I see that it is proven by Hardy in 1914 that there are an infinite number of zeros on the critical line. I also see that it was proven by Littlewood in 1921 that if $\gamma_n$ is an increasing sequence of the imaginary parts of the zeros on the critical line in the upper complex half-plane then
$$ \lim\limits_{n\to\infty}|\gamma_n-\gamma_{n-1}| =0 ~~.$$
My question is this: Is it possible that there is a greatest $\gamma_{max}$, and that there are simply an infinite number of zeros on the upper critical half-line whose imaginary parts are less than that $\gamma_{max}$?
Incidentally, I am looking for the paper where Littlewood proved the above limit but all the papers I find from 1921 are Littlewood and Hardy together, but I think this was Littlewood's sole result. If you know the paper where the above limit was proven, I am looking for that too. THANKS!!!
No. The zeros of a holomorphic function are isolated, and so there are only finitely many on any compact set contained in its domain. So in any set of the form $\{1/2+it:-T\le t\le T\}$ there are only finitely many zeros of $\zeta(s)$.