Riemann Z function, bounds on number of non-trivial zeros along horizontal lines, rather than vertical ones Concerning the non-trivial zeros of the Riemann Zeta function, one can find quite a lot of literature on:


*

*the rate of growth of the number of zeros along the vertical critical line,

*the zero-free regions of the critical strip

*bounds on the number of hypothetical non-trivial zeros inside the critical strip, but off the critical line (though none have been found as of today, and never will if the RH is true)
However, I was unable to find any work concerned with estimates of the maximum number of hypothetical non-trivial zeros which may possibly lie on a same horizontal section of the critical strip (i.e. just the horizontal lines for fixed t values).
Of course, it is well known that the functional equation implies that any hypothetical zero  (1/2-a+it) must be symmetrically mirrored on the other side of the critical line by a zero (1/2+a+it).
I imagine that such a set of hypothetical zeros would need to be discrete, as zeros of holomorphic functions are isolated, but I have no hint whatsoever about its cardinality (instinctively, we might feel that at most there are probably finitely many).
But perhaps some of you reading this question might know better.   
 A: It $t$ is not an ordinate of a zero of $\zeta(s)$, define
$$ S(t) = \frac{1}{\pi} \arg \zeta(1/2+it) = -\frac{1}{\pi} \Im \int_{1/2}^\infty \frac{\zeta'}{\zeta}(\sigma+it) d\sigma$$
and define
$$ S(t)= \lim_{\delta\to 0} \frac{1}{2}\Big(S(t+\delta) + S(t-\delta)\Big)$$
otherwise. Then the number $N(T)$ of zeros of $\zeta(s)$ in the strip $0<\Im s \le T$ is 
$$ N(T) = \frac{T}{2\pi}\log \frac{T}{2\pi e} +\frac{7}{8}+S(T)+O(\frac{1}{T})  $$
where the big-$O$ term is actually continuously differentiable. For a proof, look either in Titchmarsh's book on the zeta-function or in Montgomery & Vaughan's "Multiplicative Number Theory, I."
By continuity, the quantity you are looking for is precisely
$$ \lim_{\delta\to 0} \Big(S(t+\delta) - S(t-\delta)\Big).$$
Unconditionally, I think Tim Trudgian has the best results for this quantity showing that
$$ |S(t)| \le 0.111 \log t + 0.275 \log \log t + 2.450$$
for $t>e$ (so your quantity is bounded by essentially twice this amount). This can be sharpened if $t$ is allowed to tend to infinity.
As is mentioned in previous comments/answers, assuming the Riemann hypothesis (RH) you are looking for bounds on the multiplicity of a zero. In this case, Goldston & Gonek showed that
$$  \lim_{\delta\to 0} \Big(S(t+\delta) - S(t-\delta)\Big) \le \Big(\frac{1}{2}+o(1)\Big) \frac{\log t}{\log \log t} $$
as $t\to\infty$ using the Guinand-Weil explicit formula. 
References:
http://arxiv.org/pdf/1208.5846.pdf
http://arxiv.org/pdf/math/0511092v1.pdf
A: Best bound is O(log T) also for the multiplicity of zeros. Under RH, slightly better O(Log T/log log T). Under Lindeloeff, o(log T). This is pretty bad, because the conjecture is one.
Edit: There are slightly better bounds on the multiplicity of zeros see Ivic: arxiv.org/pdf/math/0501434
The situation is similar to that for the Selberg Zeta function. Best bound here O( T/ log T). Here, the conjecture is O(1), one for the modular group.
