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I'm trying to understand Riemann Zeta functions particularly for s=1/3

I know [tex]\zeta(s)=\sum_{n=1}^\infty\frac{1}{n^s}[/tex]

and converges for Re(s)>1

Ok, but what about for s=1/3, then

[tex]\zeta(1/3)=\sum_{n=1}^\infty\frac{1}{n^{1/3}}=

\sum_{n=1}^\infty\frac{1}{\sqrt[3]{n}}[/tex]

Theoretically it should not converge, but when I put [tex]\zeta(1/3)[/tex] on Wolfram I get approximately -0,9

May you kindly explain me this result, please?

________________________

Why? Because I'm trying to solve this problem:

[tex]\left[\frac{1}{\sqrt[3]{4}}+\frac{1}{\sqrt[3]{5}}+\frac{1}{\sqrt[3]{6}}+................+\frac{1}{\sqrt[3]{1000000}}\right][/tex]

where [tex]\left[x\right] =[/tex] is the Greatest Integer function

And I thought Riemann Zeta functions might be the solutions...

Any tips?

Thank a lot in advance guys for your support...