On the spacing of the zeros of the Riemann zeta function Suppose the Riemann zeta function has infinitely many zeros $\rho$ with $\Re(\rho)=\sigma$. Does it follow that for every large $T>0$, there exists some $t$ such that $T<t<3T$, where $t=\Im(\rho)$ ?
Existing literature seems to support this for $\sigma=1/2$.
 A: On the critical line, zeros are analyzed with the Hardy function which is real and has very good approximation formulas (Riemann-Siegel)- tons of references online including a fairly good book by Ivic on this subject (Hardy function), or a more recent one by Iwaniec, so lots of things are doable
Nothing is really known about critical strip zeros off the critical line (they are generally assumed not to exist of course) beyond some classical density theorems which are fairly weak; while there are approximation formulas for the zeta everywhere in the critical strip, they are weaker outside of the critical line as the two sums that are involved are not symmetrical and the approximation is not (essentially - up to function of constant modulus one in fact) real anymore  
A: To expand on Conrad's reference to 'some classical density theorems', these are generally called Zero Density Theorems, Titchmarsh's book is a good place to start.  See also the MathOverflow question Zeta Function: Zero Density Theorems
