What is the explanation for similar decimal digits in values of Riemann zeta function with certain arguments close to one?

Question

In Mathematica I tried these values close to one as arguments for the Riemann zeta function:

Zeta[1.000000000000010000000000000000000000000000000]
Zeta[1.000000000000020000000000000000000000000000000]
Zeta[1.000000000000040000000000000000000000000000000]
Zeta[1.000000000000080000000000000000000000000000000]
Zeta[1.000000000000160000000000000000000000000000000]
Zeta[1.000000000000320000000000000000000000000000000]
Zeta[1.000000000000640000000000000000000000000000000]
N[EulerGamma, 30]

Zeta[1.000000000000010000000000000000000000000000000^-1]    
Zeta[1.000000000000020000000000000000000000000000000^-1]
Zeta[1.000000000000040000000000000000000000000000000^-1]
Zeta[1.000000000000080000000000000000000000000000000^-1]
Zeta[1.000000000000160000000000000000000000000000000^-1]
Zeta[1.000000000000320000000000000000000000000000000^-1]
Zeta[1.000000000000640000000000000000000000000000000^-1]
N[1 - EulerGamma, 30]

And got the output:

1.000000000000005772156649015336*10^14
5.000000000000057721566490153432*10^13
2.5000000000000577215664901535773*10^13
1.2500000000000577215664901538686*10^13
6.2500000000005772156649015445111*10^12
3.12500000000057721566490155616168*10^12
1.56250000000057721566490157946275*10^12
0.577215664901532860606512090082

-1.000000000000004227843350984679*10^14
-5.000000000000042278433509846860*10^13
-2.5000000000000422784335098470052*10^13
-1.2500000000000422784335098472965*10^13
-6.2500000000004227843350984787899*10^12
-3.12500000000042278433509849044046*10^12
-1.56250000000042278433509851374153*10^12
0.422784335098467139393487909918

So in the arguments above there are the powers of two in the decimal digits, and in the output there are the digits of the Euler gamma or Euler Mascheroni constant.

What is the explanation for these similar decimal digits?

I have looked at the series expansion of the zeta function but I did not understand why.

Answer 1

There's nothing mysterious going on here, and it has nothing to do with digits or bases or powers of $2$. The Laurent series of $\zeta(z)$ at $z=1$ is

$$ \zeta(z)=\frac1{z-1}+\gamma + o(1)\;, $$

and this is exactly what you're seeing. It's not that the digits appear in some strange place; it's $\gamma$ itself, and the digits appear somewhere in a decimal expansion only because the numbers are being displayed in scientific notation.

Answer 2

Note that the "recurring digits" are actually always at the same position with respect to the real decimal point. So the numerical evidence is simply

$$\zeta(1+\varepsilon) \approx \frac{1}{\varepsilon} + \gamma $$

for small $\varepsilon$. The next term seems to be on the order of $\varepsilon$ itself.

These are indeed the first terms of the Laurent series for $\zeta(s)$ around $s=1$.