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Denoted as $\zeta(s,a)$ for a > 0

Where do I find topics on the Hurwitz zeta function for a < 0?

Any links or resources would be appreciated. (Please dont mention wiki or mathworld)

Thanks

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2 Answers 2

up vote 2 down vote accepted

Well, there's the DLMF and the Wolfram Functions site... which people really should be checking out first when they encounter an unfamiliar "special function".

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I had already checked those sites (they come up in google)... But thanks anyways :) It will be useful for people who do not know it... +1 –  Roupam Ghosh Oct 22 '10 at 3:12
    
@rpg: Well, any search for properties of Hurwitz ζ would be well-served by also looking at the more general Lerch Φ... –  J. M. Oct 22 '10 at 3:15
    
Thanks... I'll look into Lerch –  Roupam Ghosh Oct 22 '10 at 4:43

There is a very simple connection between $\zeta(s,a)$ when $a$ is negative vs when $a$ is positive.

First you need to know that $\zeta(s,a)$ is undefined when $a$ is a negative integer. This is because then $(n+a)^{-1}$ can be undefined when $n=-a.$ So you must let $a$ be not an integer.

If you do that, we can then use an identity: $\zeta(s,a)= \frac{1}{a} + \zeta(s,a+1)$ which holds by analytic continuation to $\mathbb{C}\setminus{1}.$ By using this repeatedly, we can eventually shift the zeta function to a positive value.

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