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<h1 align=center> The Density Hypothesis
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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 Lindel\"of 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 [
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<a href="http://www.ams.org/msnmain?fn=130&s1=88c:11049&pg1=MR">MR 88c:11049</a>
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Back to the 
<a href="http://www.aimath.org/WWN/rh/"> main index</a> 
for The Riemann Hypothesis.
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