I am trying to prove the following limit of a function involving the Hurwitz Zeta function:
$$ \lim_{N \to \infty} \frac{\zeta(-d, 1 + N) - \zeta(-d, 1 + p N)}{N^{1 + d}} = \frac{-1 + p^{1 + d}}{1 + d} $$
where $d$ is a positive integer, and $p$ is a real number such that $0 < p < 1$.
I have tried to prove this limit directly, but I am not sure how to handle the Hurwitz Zeta function in this context.
Any help or insights on how to approach this problem would be greatly appreciated.
We can use the identity $$\zeta(-d, a)= -\frac{B_{d+1}(a)}{d+1}, \quad (a>0, \ d \in \mathbb{N}), $$ where $B_{d}(a)$ is the $dth$ Bernoulli polynomial. This identity is typically obtained from the Hankel contour integral representation of the Hurwitz zeta function.
As $ a \to +\infty$, $B_{d+1}(a)= \sum_{k=0}^{d+1} \binom{d+1}{k}B_{d+1-k} \, a^{k}$ is asymptotic to $a^{d+1}$.
Therefore, $\zeta(-d,a) $ is asymptotic to $- \frac{a^{d+1}}{d+1}$ as $a \to + \infty$, and $$ \begin{align} \lim_{N \to \infty} \frac{\zeta(-d, 1 + N) - \zeta(-d, 1 + p N)}{N^{d+1}} &= \lim_{N \to \infty}\frac{-(1+N)^{d+1}+(1+pN)^{d+1}}{N^{d+1}(d+1)} \\ &= \lim_{N \to \infty} \frac{-N^{d+1} \left(\frac{1}{N}+1 \right)^{d+1} + N^{d+1} \left(\frac{1}{N}+p \right)^{d+1}}{N^{d+1}(d+1)} \\ &= \lim_{N \to \infty} \frac{-\left(\frac{1}{N}+1 \right)^{d+1} + \left(\frac{1}{N}+p \right)^{d+1}}{d+1} \\ &= \frac{-1+p^{d+1}}{d+1}. \end{align}$$
The limit holds for all positive values of $p$.