Riemann Z function, bounds on number of non-trivial zeros along horizontal lines, rather than vertical ones

Question

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.

Answer 1

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

Answer 2

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.