Logistic function approximation of the real valued Riemann $\zeta(x)$ function

Question

Given the function: $$f(x)=\dfrac{a}{1-b\exp(-cx)}+d$$ where: $a = 0.7071$, $b = 2.21$, $c = 0.7672$, $d = 0.2942$, I found the following inequality: $$|\zeta(x) - f(x)|\lt \epsilon$$ for $x\in\mathbb{R}$ and $x\ge2$, with $$\epsilon = 1.3\times10^{-3}$$ Among many functions, the $f(x)$ is the best for the $\zeta(x)$ approximation. Why the logistic function added to a constant can give a good approximation of the $\zeta(x)$? Thanks.

Answer 1

Using integer values $2 \leq x \leq 10$, your model and brute force, I obtain something better using $$f(x)=\frac{0.661382}{1-2.20299 e^{-0.747951 x}}+0.338967$$ which shows a better distribution of residuals and much lower errors.

You should notice that $a+d$ is almost equal to $1$. So, rerunning the model as $$f(x)=\dfrac{a}{1-b\exp(-cx)}+(1-a)$$ I obtained $$f(x)=\frac{0.643976}{1-2.19926 e^{-0.740259 x}}+0.356024 $$ which as good as the previous using one less parameter.

We can still improve the model forcing $f(2)=\frac{\pi ^2}{6}$ since the previous model gives $f(2)=1.64494$ while $\frac{\pi ^2}{6}=1.64493$. This will remove another parameter from the regression. For example, you could replace in the model $$b=\frac{\left(\pi ^2-6\right) e^{2 c}}{6 a+\pi ^2-6}$$ and stay with only two parameters $(a,c)$ to be tuned. Doing so, we obtain $$f(x)=\frac{0.644005}{1-2.19922 e^{-0.740264 x}}+0.355905 $$ which is still very good.

Added to this answer

Looking at a model $$g(x)=\frac{\zeta(2)}{\zeta(x)}=a \Big(1+b e^{-c(x-2)}\Big)$$ and imposing $g(2)=1$ and $g(\infty)=\frac{\pi ^2}{6}$ leads to $$a=\frac{\pi ^2}{6}$$ $$b=\frac{6}{\pi ^2}-1$$ Imposing now that $g'(2)$ be exact leads to $$c=-\frac{36 \zeta '(2)}{\pi ^2 \left(\pi ^2-6\right)}=0.883751$$ whcih is not bad at all without any adjustment. Letting $c$ to be adjustable and using now the model $$f(x)=\frac{1}{1+\left(\frac{6}{\pi ^2}-1\right) e^{-c (x-2)}}$$ to approximate $\zeta(x)$, the optimum value is $c=0.831036$ and the fit is really good at the price of a single tunable parameter.