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.

• Perhaps I am missing something, but since these zeros would all have to lie in the critical strip, their real paerts are bounded, if you now fix the imaginary part, you have a bounded set. So any inifinite set would have an accumaltion point.
– user9072
Mar 26 '13 at 18:07
• I think that from standard asymptotics on the number of zeros of height up to T you get a bound of O(log T). Apparently even on RH is is only known that the multiplicity of 1/2 + iT as a zero is O(log T/log log T). So I guess you shouldn't expect too much better than this O(log T) bound, Mar 26 '13 at 18:27
• The number of (nontrivial) zeros with imaginary part between $0$ and $T$ is $\frac T{2\pi}\log\frac T{2\pi}-\frac T{2\pi}+O(\log T)$, so the number of zeros with imaginary part in $[T,T+1)$ is $O(\log T)$, with average about $\frac1{2\pi}\log T$. This includes roots both on and off the critical line. Mar 26 '13 at 18:31

2 Answers

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

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.

• I feel that bounds for the multiplicity of zeros is a more reasonable question, because we expect RH to be true. Mar 26 '13 at 19:25
• Micah, thank you for those references. So, it appears that the consensus is that, for a horizontal line at height T, no stricter bounds are available than those already known for the number of zeros inside a rectangle corresponding to the critical strip up to height T. But thanks for pointing out that that is already sufficient to at least show that along said horizontal lines there cannot lie more than at most finitely many zeros (I had missed that now obvious implication ...).
– Luca
Mar 27 '13 at 8:43
• and whose number cannot exceed said already known limits you recalled. It is anyway a better estimate than simply relying on the fact that the zero set of a non-constant holomorphic function on a connected open set is discrete.
– Luca
Apr 8 '13 at 8:00