Turan showed that if for all sufficiently large , the th partial sum of does not vanish in then the Riemann Hypothesis follows.

He [
MR 10,286a] strengthened this criterion
by showing that for every
there is an such that if the th partial sum

of the zeta-function has no zeros in for all then the Riemann Hypothesis holds.

H. Montgomery [
MR 87a:11081] proved that this approach cannot work because
for any positive number
the th partial sum of has zeros in the half-plane

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