analytic continuation of Riemann zeta (using integral)
When , this series converges; furthermore, this convergence is uniform on compact subsets of this half-plane, hence the series converges to an analytic function on this half plane. However, the series diverges when we have , so this series cannot be used to define the zeta function in the whole complex plane, which is why we must make an analytic continuation.
To make this continuation, we start by changing the variable in an integration:
This provides us with an integral representation of our summand. Substituting this into the series, we find that
We note that
because the series converges, hence it is possible to interchange integration and summation and subsequently sum a geometric series.
As it stands, the integral representation we have is not of much use for analytically continuing the zeta function because the integral diverges when on account of the fact that the integrand behaves like when is close to zero. However, it is possible to make use of the theorem of Cauchy to move the path of integration away from zero.
Given a real number , define the contour on the Riemann surface of as follows: passes from to along a lift of the real axis, then continues along the circle of radius clockwise, and finally goes from to .
We now examine the integral over such a contour by breaking it into three pieces.
We may estimate the third integral in absolute value like so:
The expression represents an analytic function of , and hence a bounded function of in a neighborhood of . When , it happens that , so
The third integral differs from the first integral by a phase, so they may be combined by pulling out this common factor. When , we may take the limit as approaches after doing so to obtain the following:
whenever and , which trivially implies that
for any between and . Therefore,
when . This integral converges for all complex because the exponential grows more rapidly than the power. Furthermore, this integral defines an analytic function of , so we have an analytic continuation of the zeta function to the whole complex plane minus the point 1.
|Title||analytic continuation of Riemann zeta (using integral)|
|Date of creation||2013-03-22 16:53:59|
|Last modified on||2013-03-22 16:53:59|
|Last modified by||rspuzio (6075)|