Why are $\lfloor\zeta(\zeta(n))\rfloor$,$\left\lfloor\frac{1}{\zeta(n)-1}\right\rceil$, and $\lceil\Gamma(\zeta(n)-1)\rceil$ so similar?

I was first investigating the sequence (A111000) defined by the function $$a_n = \lfloor\zeta(\zeta(n))\rfloor$$ where $\zeta(n)$ is the Riemann Zeta Function and $$\lfloor x \rfloor = \max\{m \in \mathbb{Z} \mid m \leq x\}$$ also known as the floor function. The sequence begins, $a_n = 2, 5, 12, 27, 58, 120, ... \text{ for } n = 2, 3, 4, 5, 6, 7, ...$

I then tried investigating other functions relating to the Zeta Function that would satisfy the requirement that the positive infinite limit would tend toward infinity. This led me to $$b_n = \left\lfloor\frac{1}{\zeta(n)-1}\right\rceil$$ where $\lfloor x \rceil = \left\lfloor x + \frac{1}{2} \right\rfloor$. And secondly, $$c_n = \lceil\Gamma(\zeta(n)-1)\rceil$$ where $\Gamma(n)$ is the Gamma Function and $$\lceil x \rceil = \min\{n \in \mathbb{Z} \mid n \geq x\}$$ I found that for some reason, the generated sequences were very similar. $$a_n = 2, 5, 12, 27, 58, 120, 245, 498, 1006, 2024, 4064, 8149, 16327, 32692, ...$$ $$b_n = 2, 5, 12, 27, 58, 120, 245, 498, 1005, 2024, 4064, 8149, 16327, 32692, ...$$ $$c_n = 2, 5, 12, 27, 58, 120, 245, 498, 1005, 2023, 4064, 8149, 16327, 32692, ...$$ The seemed to be the same for almost all $n$. I tested this up to $n = 1000$ and found the following observations to be true up to that point. $$a_n \geq b_n \geq c_n \text{, and by Squeeze Theorem, if } a_n = c_n \text{ then } a_n = b_n = c_n$$ $$a_n - c_n \leq 1$$ I am curious why these values are so similar. Is there some equality or near equality I do not know about. Is it possible to prove my observations? How?

• You should expect the first two to be close; $\zeta(n)$ will be very close to $1$ as $n\to\infty$ (in fact, $\approx 1+\frac1{2^n}$, so it converges pretty quickly to $1$!), and it's a classical result that $\zeta(1+\epsilon) \approx \epsilon^{-1}$, which is really the core content of $a_n\approx b_n$. – Steven Stadnicki May 1 '18 at 6:48

The closeness of these terms follows form the Stieltjes series expansion of the Riemann Zeta function and the series expansion of the gamma fucntion. We have

$$\zeta(s) = \frac{1}{s-1} + \sum_{n=0}^{\infty}\frac{(-1)^n}{n!}\gamma_n(s-1)^n$$

where $\gamma_n$ are the Stieltjes constants.