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I was reading about some of the zero density Theorems for my Analytic Number Theory Topics course. While looking over some more complicated results and proofs a few simple questions came up:

First, let $$N(\sigma,T)=|\{ \rho=\beta+i \gamma \text{ }:\text{ } \zeta(\rho)=0, \text{ } 0< \gamma < T, \text{ } \sigma\leq\beta<1 \}|$$ be the cardinality of the set of zeros of the zeta function real part greater than $\sigma$ and imaginary part between 0 and $T$.

We call theorems pertaining to the size of $N(\sigma,T)$ zero density theorems. (Complicated bounds on the size of the empty set) These estimates are usually written in the form $N(\sigma , T)\ll T^{A(\sigma)(1-\sigma)+\epsilon}$ where the $\ll$-constant is uniform in $\sigma$. (So the smaller $A(\sigma)$ is, the better, and all such theorems concern the size of $A(\sigma)$)

My questions are:

1) Why do we have a factor of $(1-\sigma)$ in the exponent $T^{A(\sigma)(1-\sigma)}$? It is clear to me what this implies about the density of the zeros, but why does it arise so naturally?

2) The so called density hypothesis is that $A(\sigma) \leq 2$. The best known bound is $A(\sigma) \leq 2.4$ (or possibly 2.3) What is so special about $A(\sigma)\leq 2$? Are there links between this value and certain theorems one may hope to prove? Why does this in particular deserve such a name as "the density hypothesis?"

Thanks a lot!!

(Also, if you have a good reference book or paper that talks about these issues I would be happy to know about it!)

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I fixed what I presume was the problem with the TeX –  David Roberts Jan 20 '11 at 4:18
Don't have it handy, but Iwaniec and Kowalski's Analytic Number Theory has a pretty thorough discussion. –  Frank Thorne Jan 20 '11 at 4:33
Thanks for the editing help, I'll remember it in the future. –  Eric Naslund Jan 20 '11 at 5:37
I am far from expert on this topic, but since one can show that there are no zeroes on the line $\sigma=1$, and currently cannot find any stronger uniform upper bound on the real parts of the zeroes in the critical strip, it doesn't surprise me that all known estimates produce an exponent of the form $1−\sigma$. –  Emerton Jan 20 '11 at 6:48
A good treatment of zero-density theorems can be found in Aleksandar Ivic's book "The Riemann zeta-function. The theory of the Riemann zeta-function with applications", chapter 11. It should be noted that the best range (in $\sigma$) where the density hypothesis is known to hold is $\sigma \geq 25/32$ by Bourgain, which is an improvement of the result $\sigma \geq 11/14$ of Jutila from the seventies. –  Johan Andersson Jan 20 '11 at 14:53

2 Answers 2

up vote 8 down vote accepted

If you state the Density Hypothesis as $$ N(\sigma,T) \ll T^{ 2(1-\sigma)+\epsilon}$$ the factor $1-\sigma$ seems natural because the exponent $2(1-\sigma)$ is 1 when $\sigma=1/2$, where there are $ \gg T\log T$ zeros, and is $2(1-\sigma)$ is 0 when $\sigma=1$ where there are no zeros. So as a linear function of $\sigma$, the exponent is best possible.

The Density Hypothesis is a named hypothesis because it can be used to derive interesting results. For instance, if you let $p_n$ denote the $n$th prime, the Density Hypothesis implies that $$ p_{n+1}-p_n \ll p_n^{1/2+\epsilon}.$$ This is almost as strong as what can be proved under the assumption of the Riemann Hypothesis.

Remark: It is known that Riemann Hypothesis $\implies$ Lindelof Hypothesis $\implies$ Density Hypothesis, but none of the reverse implications have been proved.

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In their book Analytic Number Theory, Iwaniec and Kowalski state that the exponent of $2(1-\sigma)$ is conjectured "by analogy to the convexity principle concerning the order of $\zeta(s)$ on vertical lines." –  Micah Milinovich Jan 20 '11 at 5:12

To resonate with John's answer, it is in the application where exponents of the form $c(1-\sigma)$, with $c$ constant, are natural and important. I recommend you read page 265 of Iwaniec-Kowalski: Analytic number theory, after which you will see this connection very clearly. Let me refine this answer.

For $\sigma$ close to $1/2$ Ingham (Quart. J. Math. 8 (1937), 255-266) proved that the exponent $2(1-\sigma)+\epsilon$ follows from the Lindelöf Hypothesis, so the Density Hypothesis was born. On the other hand, Turán (Acta Math. Hung. 5 (1954), 145-163) conjectured that for $\sigma\geq 1/2+\delta$ it should be possible to derive from the Lindelöf Hypothesis the much stronger exponent $\epsilon$. He accomplished this derivation for $\sigma\geq 3/4+\delta$ in a joint paper with Halász (J. Number Theory 1 (1969), 121–137.). In the same paper they also proved unconditionally that $(1-\sigma)^{3/2}\log^3(1-\sigma)^{-1}$ is an admissible exponent when $1-\sigma$ is sufficiently small.

This shows that, from the point of our current understanding, the dependence $1-\sigma$ is rather natural when $\sigma$ is close to $1/2$, but less so when $\sigma$ is close to $1$. There is a definite turning point at $\sigma=3/4$: if one is very-very optimistic, current technology (Halász' inequality and their refinements due to Montgomery, Huxley, Jutila, Bourgain and others) might lead to a proof of the Density Hypothesis for $\sigma>3/4$, but certainly some very new ideas will be needed to make an improvement for $\sigma\leq 3/4$.

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