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The zeta regularization of $prod_{m=-infty}^infty (km+u)$ Answer

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The zeta regularization of $prod_{m=-infty}^infty (km+u)$

Background: I’m dealing with the computation of the zeta regularization of the innumerable product given by

$$prod_{m=-infty}^infty (km+u)$$

for a actual constructive $ok$ and $Im(u)neq 0$. From J. R. Quine, S. H. Heydari and R. Y. Song (Example 10) I do know that the zeta regularization of $prod_{m=-infty}^infty (m+u)$ is given by

$$prod_{m=-infty}^infty (m+u) = commence{instances}1-e^{2pi i u} & Im(u) > 0 1-e^{-2pi i u} & Im(u) < 0end{instances}$$

so a workable reasoning is as follows:

$$prod_{m=-infty}^infty (km+u) = prod_{m=-infty}^infty ok(m+uk^{-1})$$

Now, utilizing system (1) from J. R. Quine, S. H. Heydari and R. Y. Song we have now that

$$prod_{m=-infty}^infty ok(m+uk^{-1}) = ok^{Z(0)}prod_{m=-infty}^infty (m+uk^{-1})$$

and we’re decreased to computing $Z(0)$ the place $Z(s)$ is the analytic prolongation of $Z(s)=sum_{m in mathbb{Z}}(m+uk^{-1})^{-s}$. We can write $Z(s)$ as

$$Z(s) = sum_{m > 0}(m+uk^{-1})^{-s}+(uk^{-1})^{-s}+sum_{m <0}(m+uk^{-1})^{-s}$$

By including and subtracting the $m=0$ time period to each of the succession and by altering variable from $m$ to $-m$ within the sum listed by traverse integers we get

$$commence{align}
Z(s) &= sum_{m=0}^infty(m+uk^{-1})^{-s} – (uk^{-1})^{-s} + sum_{m=0}^infty(uk^{-1}-m)^{-s}
&= sum_{m=0}^infty(m+uk^{-1})^{-s} – (uk^{-1})^{-s} + (-1)^{-s}sum_{m=0}^infty(m-uk^{-1})^{-s}
&= zeta(s,uk^{-1}) -(uk^{-1})^{-s} + (-1)^{-s}zeta(s,-uk^{-1}) aim{align}$$

the place $zeta(s,a)$ is the Hurwitz Zeta duty. So that by utilizing system 25.11.13 once more in https://dlmf.nist.gov/25.11#E13 we create that

$$Z(0) = dfrac{1}{2}+uk^{-1}-1+dfrac{1}{2}-uk^{-1}=0$$

and we aim up with

$$prod_{m=-infty}^infty ok(m+uk^{-1}) = ok^{Z(0)}prod_{m=-infty}^infty (m+uk^{-1}) = prod_{m=-infty}^infty (m+uk^{-1}) = commence{instances}1-e^{2pi i uk^{-1}} & Im(uk^{-1}) > 0 1-e^{-2pi i uk^{-1}} & Im(uk^{-1}) < 0end{instances}$$

Question: I’m not positive sufficient with these manipulations moving analytic prolongations so I could breathe grievance within the above derivation of $Z(0)=0$, Is the above outcome rectify? Does it stay genuine if one replaces the situation $ok$ actual and constructive with $ok$ a nonzero complicated quantity?

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