Zeta Function: Zero Density Theorems.

Question

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!)

Answer 1

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.

Answer 2

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 $\sigma\leq 3/4$.