Riemann zeta function has no zeros on
This article shows that the Riemann zeta function has no zeros along the lines or . That implies that all nontrivial zeros of lie strictly within the critical strip . As the article points out, this is known to be equivalent to one version of the prime number theorem.
It can in fact be shown that for any with if
for some constant . By using the functional equation
we have also that if
Bounding the zeros of away from , leads to a version of the prime number theorem with more precise error terms.
Proof. Notice that for
If , then , so that
Using equation (1), we then have
But if has a zero at , then
since the first factor gives a pole (http://planetmath.org/Pole) of order 3 at and the second factor gives a zero of order at least 4 at . This contradicts equation (2).
Proof. Use the functional equation
and set . The theorem implies that the RHS is nonzero, so the LHS is as well. Thus .
|Title||Riemann zeta function has no zeros on|
|Date of creation||2013-03-22 17:54:37|
|Last modified on||2013-03-22 17:54:37|
|Last modified by||rm50 (10146)|