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let's define $p_\alpha(n) = \displaystyle\int_1^n x^\alpha dx$ so that $\left\{\begin{array}{lll} p_0(n) &=& n-1 \\ p_{-1}(n) &=& \ln n \\ p_\alpha(n) &=& \frac{\textstyle n^{\alpha+1} - 1}{\alpha+1} \quad \text{for} \ \ \ \alpha \ne -1\end{array}\right.$

Then, take $p_\alpha(n)$ to make a Dirac comb and look at its Laplace transform : $$P_\alpha(s) = LT\left\{ \sum_{n=1}^\infty \delta\left(t - p_\alpha(n)\right)\right\} = \sum_{n=1}^\infty e^{\textstyle-s \,p_\alpha(n)}$$

We get that $\displaystyle P_0(s) = \sum_{n=1}^\infty e^{-s (n-1)} = \frac{1}{1-e^{-s}}$ and $\displaystyle P_{-1}(s) = \sum_{n=1}^\infty n^{-s} = \zeta(s)$.

Question : what do we know about the zeros and poles localization of the analytic continuation of $P_\alpha(s)$ for other values of $\alpha$ in $[-1;\infty[$ ?

Are there numerical evidences that they are always aligned on a few lines of the complex plane ?

Are these localization at least continuous with respect to slow variations of $\alpha$ ?

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