Hyperzeta function

[Balarka Sen]

I invented a function, somewhat similar to that of the Riemann zeta, but replacing the reciprocal of powers by superexponentiation. Namely,

\( \mathfrak{H}(z) = \sum_{n=1}^{\infty} \frac{1}{{}^z n} \) where by definition \( {}^z 1 = 1 \)

This seems to diverge for z < 1.

I tried many methods of analytically continue it in the complex plane analogues to zeta but none of them seems to be that useful. But Cohen-Villegas-Zagier's acceleration (CVZ) method seems to be converging it in a unique way. But I haven't dared to prove whether CVZ gives the right or even an analytic continuation.

There are many interesting properties I found under CVZ like : H^[m](z) seems to be converging towards either -7.0744329865020 + i39.959874355410 or -7.0744329865020 - i39.959874355410. If we denote these values by z1 and z2, it seems that H(z1) = z2. And also, interestingly, z1 = z2* ! These are surely not fixed points but what are they?

I've also posted this in a forum : http://www.mymathforum.com/viewtopic.php?f=15&t=38901

I've added a little details there so please read it. (I am too lazy to include those detailed informations here right now)

Any comments and graphs would be appreciated,
Balarka
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