analytic continuation of Riemann zeta to critical strip
The (see the general power) of the series
defining the Riemann zeta function for , are holomorphic in the whole -plane and the series converges uniformly in any closed disc of the half-plane ((let with and ; then for a positive for all ; the series converges since ; thus the series (1) converges uniformly in the closed half-plane , by the Weierstrass criterion (http://planetmath.org/WeierstrassCriterionOfUniformConvergence))). Therefore we can infer (see theorems on complex function series (http://planetmath.org/TheoremsOnComplexFunctionSeries)) that the sum of (1) is holomorphic in the domain .
We use also the fact that the series
If we multiply the series (1) by the difference , every other of the series changes its sign and we get the series (2). So we can write
which is valid when the denominator does not vanish and . The zeros of the denominator are obtained from , i.e. from
Thus the zeros of the denominator of (3) are on the line .
Now the function on the right hand side of (3) is holomorphic in the set
and the values of this function coincide with the values of zeta function in the half-plane .
This result means that, via the equation (3), the zeta function has been analytically continued (http://planetmath.org/AnalyticContinuation) to the domain , as far as to the imaginary axis.
Remark. In reality, all points (4) except are removable singularities of given by (3), due to the fact that they are also zeros of . The fact is considered in the entry zeros of Dirichlet eta function.
Charles Hermite has shown that the zeta function may be analytically continued to the whole -plane except for a simple pole at , by using the equation
See this article (http://planetmath.org/AnalyticContinuationOfRiemannZetaUsingIntegral).
|Title||analytic continuation of Riemann zeta to critical strip|
|Date of creation||2015-08-22 13:33:33|
|Last modified on||2015-08-22 13:33:33|
|Last modified by||pahio (2872)|
|Defines||alternating zeta function|