Let $N(\sigma,T)$ denote the number of zeros $\rho=\beta+\gamma i$ of the Riemann zeta function satisfying $\beta\ge \sigma$ and $0<\gamma\le T$, counted with multiplicity. Then the "Density Hypothesis" usually means one of the two following unproven conjectures.

1."Weak" Density Hypothesis: Let $\varepsilon>0$ be fixed. Then $$N(\sigma,T)=O(T^{2(1-\sigma)+\varepsilon})$$ uniformly in $1/2\le \sigma\le 1$ as $T\to +\infty$.

2. "Strong" Density Hypothesis: There exists a real number $\alpha\ge 1$, such that $$N(\sigma,T)=O(T^{2(1-\sigma)}(\log T)^{\alpha})$$ uniformly in $1/2\le \sigma\le 1$ as $T\to +\infty$.

From the work of Hoheisel, it is known that if the (Strong) density hypothesis is true, then \begin{align} \psi(x+x^{\vartheta})-\psi(x)\sim&\;x^{\vartheta} &&(1)\\[1mm] \pi(x+x^{\vartheta})-\pi(x)\sim&\; \min(1,\vartheta^{-1})\frac{x^{\vartheta}}{\log x} &&(2)\\[1mm] p_{n+1}-p_{n}=&\;O(p_n^{\vartheta}) &&(3) \end{align} all hold for any fixed $\vartheta>1/2$.

Question: Are there any other (at least remotely significant) consequences that would follow if one of the above density hypotheses could be proven (without assuming the Lindelöf or Riemann Hypotheses)? $(1)-(3)$ surely cannot be an exhaustive list.

Some background: The density hypothesis arose from A.E. Ingham's paper On the difference between consecutive primes (1937), where he showed that, if $\zeta(\tfrac{1}{2}+ti)=O(t^{c})$ as $t\to +\infty$, for some fixed $c>0$, then $$N(\sigma,T)=O(T^{2(1+2c)(1-\sigma)}(\log T)^{5})$$ uniformly in $1/2\le \sigma \le 1$ as $T\to +\infty$.

Thus, if the Lindelöf hypothesis is true, then we may take $c=\varepsilon$ and conclude that the weak density hypothesis is true. As explained in this post, the linear term $2(1-\sigma)$ is the best possible (or, at the very least, naturally occuring). As argued in the same post, the density hypothesis has been proven unconditionally for $\sigma \ge 25/32$ by Bourgain, although the conclusion over this range is somewhat stronger than the density hypothesis itself.

Supposing for the moment that we could in fact prove that the density hypothesis is true, then what would we gain from it?

In Ivić's book, it sais on p. 47 that "Estimates of $N(\sigma,T)$" play an important role in many applications of zeta-function theory, and for some applications concerning primes the reader is referred to chapter 12." Now $(1)-(3)$ are basically the results he is referring to. So where else are estimates of $N(\sigma,T)$ important to zeta-function theory, especially if the density hypothesis is true?


1 Answer 1


The "strong" density hypothesis implies that if $\epsilon>0$ fixed and $X$ is large, then the Lebesgue measure of the $x\in[0,X]$ such that the interval $[x,x+x^{\epsilon}]$ does not contain a prime is $o(X)$. (See the bottom of page 131 in Montgomery's "Topics in Multiplicative Number Theory".) For comparison, RH implies that one may replace $x^{\epsilon}$ with $f(x)(\log x)^2$, where $f(x)$ monotonically increases to infinity as $x\to\infty$ (proved in Selberg's paper "On the normal density of primes in small intervals, and the difference between consecutive primes", which can be found in his collected works). The threshold $x^{\epsilon}$ is interesting because Maier's matrix method shows that an asymptotic prime number theorem for $\pi(x+(\log x)^A)-\pi(x)$ cannot hold for any fixed $A>2$, so such almost-all results are the best for which one could hope.

But the density hypothesis is not the only meaningful bound on $N(\sigma,T)$. There is also Selberg's unconditional estimate

$$N(\sigma,T)\ll T^{1-\frac{1}{4}(\sigma-\frac{1}{2})}\log T,$$

which is a key tool in the first proofs of Selberg's central limit theorems for $\log|\zeta(\frac{1}{2}+it)|$ and $\arg\zeta(\frac{1}{2}+it)$. (See here.) These central limit theorems provide inspiration for conjectures on how the moments of $\zeta(s)$ grow, and they even suggest how one might possibly go about proving such conjectures. Perhaps the definitive example of this is due to Soundararajan. Even though Soundararajan's results rely on RH, the inspiration from Selberg is clear in the proofs.

Ford and Zaharescu used Selberg's estimate to study the limiting distribution of the fractional parts of the imaginary parts of the nontrivial zeros of $\zeta(s)$. As Ford, Soundararajan, and Zaharescu show (perhaps unexpectedly), this distribution is closely connected with the pair correlation of zeros of $\zeta(s)$ and the distribution of primes in short intervals (this is interesting because Selberg's density estimate is sharp near $\mathrm{Re}(s) = \frac{1}{2}$, but not near $\mathrm{Re}(s) =1$, which is the purview of the density hypothesis).

Selberg's estimate has other nice applications, but these come to mind easily.

  • $\begingroup$ Thank you for the answer. Could you provide me a reference to the first two results you mention (the prime gap statements) ? : ) I will, however, not accept the answer yet, since I believe there may be more "folklore/unwritten" results under the density hypothesis. But I will look into your answer in the meantime. $\endgroup$
    – AfterMath
    Oct 6, 2022 at 10:34
  • 1
    $\begingroup$ @AfterMath References added. $\endgroup$
    – 2734364041
    Oct 11, 2022 at 8:10

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