Approximate functional equation for the Riemann zeta function

Question

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 $2\zeta(s)$.

Answer 1

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".