Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question
1  
Funny -- I knew after reading half the question that it was you asking it :-) How do you always come up with these questions where one wonders how you ever got the idea of trying this out without already knowing why it would yield something interesting? :-) –  joriki Oct 14 '12 at 14:31
add comment

2 Answers

up vote 5 down vote accepted

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.

share|improve this answer
add comment

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 $s=1$.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.