# Approximate functional equation for the Riemann zeta function

The Riemann zeta function admits the approximation $$\zeta(s)\sim\sum_{n=1}^N\frac{1}{n^s}+\gamma(1-s)\sum_{n=1}^M\frac{1}{n^{1-s}},$$ in the critical strip, which is known as the approximate functional equation for the Riemann zeta function. Here $$\gamma$$ is the multiplier from the functional equation $$\zeta(s)=\gamma(1-s)\zeta(1-s)$$. However, the both sums seem to tend to $$\zeta(s)$$ as $$N, M\to\infty$$. I would like to get an explanation why we do not have a duplication and the sum of the two series equals $$\zeta(s)$$ and not $$2\zeta(s)$$.

## 1 Answer

For $$\;s:=\sigma+it\;$$ with $$\,\sigma\in(0,1)\;$$ Hardy and Littlewood's approximate functional equation states that : $$\tag{1}\zeta(s)=\sum_{n=1}^N\frac{1}{n^s}+\gamma(1-s)\sum_{n=1}^M\frac{1}{n^{1-s}}+O\left(N^{-\sigma}\right)+O\left(|t|^{\frac 12-\sigma}\,M^{\sigma-1}\right)$$ $$\tag{2}\gamma(s):=\pi^{1/2-s}\frac{\Gamma(s/2)}{\Gamma((1-s)/2)}\;$$ with the hypothesis $$\;|t|\approx 2\pi\,N\,M\;$$ (see page $$4$$ of Gourdon and Sebah's paper "Numerical evaluation of the Riemann Zeta-function" for details).

But this implies that nor $$N$$ nor $$M$$ are going to $$\,+\infty\;$$ ... especially since both series would be divergent at the limit!.

In fact we usually suppose $$\;N=M=\left\lfloor\sqrt{\dfrac {|t|}{2\pi}}\right\rfloor\;$$ and with the precise remainder in $$(1)$$ considered as an asymptotic expansion in $$N$$, obtain the Riemann-Siegel formula.
This remainder is not easy to evaluate and we will follow up with $$\,\sigma=\dfrac 12$$.

For $$\;s=\frac 12+it\;$$ the $$\;\gamma(1-s)\,$$ factor verifies $$\;|\gamma(1-s)|=1\,$$ but with a phase factor that we can't neglect while the sum at the right will simply be the complex conjugate of the first sum.

The interest of the Riemann-Siegel formula (and the approximate functional equation) is that alternative evaluations of $$\,\zeta(s)\,$$ using the finite sum $$\;\displaystyle\sum_{n=1}^X\frac{1}{n^s}\;$$ like Euler-Maclaurin seem to impose $$\,X\,$$ to be larger than $$\dfrac{|t|}{2\pi}$$ to be precise (as illustrated in this answer).

The Riemann-Siegel formula allows to replace this sum of $$X=\left[\dfrac{|t|}{2\pi}\right]$$ terms with the sum restrained to the $$\left[\sqrt{X}\right]$$ first terms added to the sum of the $$\left[\sqrt{X}\right]$$ first terms terms of $$\zeta(1-s)$$ multiplied by $$\,\gamma(1-s)$$ : computing $$[2\times]\,10^5$$ terms instead of $$\,10^{10}$$ makes a difference when we search large zeros! (Riemann-Siegel versus Euler-Maclaurin is described here).

The links should help you more as well as this nice paper by Carl Erickson's "A Geometric Perspective on the Riemann Zeta Function's Partial Sums".