Is Lerch's transcendent a multi-valued function? Lerch’s Transcendent is defined by $${\Phi\left(z,s,a\right)=\sum_{n=0}^{\infty}\frac{z^{n}}{(a+n)^{s}}},$$ when $|z|<1$ or $\Re s>1,|z|=1$. If $s$ is not an integer then $|\mathrm{ph}(a)|<\pi$; if $s$ is a positive integer then $a \ne 0,−1,−2,\dots$; if s is a non-positive integer then a can be any complex number. For other values of $z$, $\Phi(z,s,a)$ is defined by analytic continuation. My question is, is Lerch's Transcendent considered a multi-valued function? I guess it depends on how the term $(a+n)^s$ is interpreted. For example if $s = 3/2$ and $a = 1$. We can take $(a+n)^s$ to be either the positive root or the negative root, which will give different value to the sum. What is the traditional interpretation of such definitions in complex analysis?

Suppose the function of the variable $z$ is $\sum_{n \geq 0} z^n/(n + 1) = -\ln(1 - z)/z, \, |z| < 1$, where $\ln$ is the principal value of the logarithm. Looping once around $z = 1$ gives $-\ln(1 - z)/z \pm 2 \pi i/z$. $\Phi(z, 1, 1)$ is multi-valued in this sense.