analytic continuation of Riemann zeta to critical strip

The   $\frac{1}{n^s}=e^{-s\log n}$  (see the general power) of the series

${\displaystyle \sum _{n=1}^{\mathrm{\infty }}}{\displaystyle \frac{1}{n^s}}=\mathrm{\hspace{0.33em}1}+{\displaystyle \frac{1}{2^s}}+{\displaystyle \frac{1}{3^s}}+{\displaystyle \frac{1}{4^s}}+\mathrm{\dots },$ (1)

defining the Riemann zeta function $\zeta (s)$ for  $\mathrm{\Re }s>1$,  are holomorphic in the whole $s$-plane and the series converges uniformly in any closed disc of the half-plane  $\mathrm{\Re }s>1$ ((let  $s=\sigma +it$  with $\sigma ,t\in ℝ$  and $\sigma >1$;  then $|\frac{1}{n^s}|=\frac{1}{n^{\sigma }}\le \frac{1}{n^{1+d}}$  for a positive $d$ for all  $n=1,\mathrm{\hspace{0.17em}2},\mathrm{\dots }$;  the series  ${\sum }_{n=1}^{\mathrm{\infty }}\frac{1}{n^{1+d}}$ converges since  $1+d>1$;  thus the series (1) converges uniformly in the closed half-plane  $\mathrm{\Re }s\ge 1+d$,  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 $\zeta (s)$ of (1) is holomorphic in the domain  $\mathrm{\Re }s>1$.

We use also the fact that the series

${\displaystyle \sum _{n=1}^{\mathrm{\infty }}}{\displaystyle \frac{{(-1)}^{n-1}}{n^s}}=\mathrm{\hspace{0.33em}1}-{\displaystyle \frac{1}{2^s}}+{\displaystyle \frac{1}{3^s}}-{\displaystyle \frac{1}{4^s}}+-\mathrm{\dots }$ (2)

defining the Dirichlet eta function $\eta (s)$, a.k.a. the alternating zeta function, is convergent for  $\mathrm{\Re }s>0$  and its sum is holomorphic in this half-plane.

If we multiply the series (1) by the difference $1-{\displaystyle \frac{2}{2^s}}$, every other of the series changes its sign and we get the series (2).  So we can write

$\zeta (s)={\displaystyle \frac{\eta (s)}{1-\frac{2}{2^s}}},$ (3)

which is valid when the denominator does not vanish and  $\mathrm{\Re }s>1$.  The zeros of the denominator are obtained from  $2^s=2$,  i.e. from

$$e^{s\log 2}=e^{\log 2}.$$

This gives  $s\log 2-\log 2=n\cdot 2i\pi$ (see the periodicity of exponential function), i.e.

$s=\mathrm{\hspace{0.33em}1}+n\cdot {\displaystyle \frac{2\pi i}{\ln 2}}\mathit{\hspace{1em}}(n\in \mathbb{Z}).$ (4)

Thus the zeros of the denominator of (3) are on the line  $\mathrm{\Re }s=1$.

Now the function on the right hand side of (3) is holomorphic in the set

$$D:=\{s\in ℂ\mathrm{⋮}\mathrm{\Re }s>0\}\setminus \{1+n\cdot \frac{2\pi i}{\ln 2}\mathrm{⋮}n\in \mathbb{Z}\}$$

and the values of this function coincide with the values of zeta function in the half-plane  $\mathrm{\Re }s>1$.

This result means that, via the equation (3), the zeta function has been analytically continued (http://planetmath.org/AnalyticContinuation) to the domain $D$, as far as to the imaginary axis.

Remark.  In reality, all points (4) except  $s=1$  are removable singularities of $\zeta (s)$ given by (3), due to the fact that they are also zeros of $\eta (s)$.  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 $s$-plane except for a simple pole at  $s=1$,  by using the equation

$\zeta (s)={\displaystyle \frac{1}{\mathrm{\Gamma }(s)}}{\displaystyle {\int }_0^{\mathrm{\infty }}}{\displaystyle \frac{x^{s-1}}{e^x-1}}𝑑x.$ (5)

See this article (http://planetmath.org/AnalyticContinuationOfRiemannZetaUsingIntegral).

Title	analytic continuation of Riemann zeta to critical strip
Canonical name	AnalyticContinuationOfRiemannZetaToCriticalStrip
Date of creation	2015-08-22 13:33:33
Last modified on	2015-08-22 13:33:33
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	23
Author	pahio (2872)
Entry type	Example
Classification	msc 30D30
Classification	msc 30B40
Classification	msc 11M41
Related topic	RiemannZetaFunction
Related topic	AnalyticContinuation
Related topic	MeromorphicExtension
Related topic	CriticalStrip
Related topic	AnalyticContinuationOfRiemannZetaUsingIntegral
Related topic	FormulaeForZetaInTheCriticalStrip
Related topic	GammaFunction
Defines	alternating zeta function