The Density Hypothesis

The Density Hypothesis is the assertion

$$N(\sigma,T) = O(T^{2(1-\sigma)+\varepsilon})$$

for all $\varepsilon>0$. Note that this is nontrivial only when $\sigma > \frac12$.

The Density Hypothesis follows from the Lindelöf Hypothesis. The importance of the Density Hypothesis is that, in terms of bounding the gaps between consecutive primes, the density hypothesis appears to be as strong as the Riemann Hypothesis.

Results on $N(\sigma,T)$ are generally obtained from mean values of the zeta-function. Further progress in this direction, particularly for $\sigma$ close to $\frac12$, appears to be hampered by the great difficulty in estimating the moments of the zeta-function on the critical line.

See Titchmarsh [ MR 88c:11049], Chapter 9, for an extensive discussion.

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