I would like to know if is it possible to prove or disprove the following conjecture: Given the following limit: $$L(x,N)=\lim_{N\to\infty}\left(H^{(-x)}_N-NH_N\right)$$ we have: $L(x,N)\lt+\infty$ for a value of $x\in\mathbb{R}$ such that $1.3\lt x\lt1.5$
This would mean to prove that it exists an $x$ bounded as before such that, the quantity: $$\Phi(x,N)=\dfrac{1}{N}\left(-\zeta(-x,N+1)+\zeta(-x)\right)-N\left(\gamma+\Psi^{(0)}(N+1)\right)$$ is bounded for $N\to+\infty$. This because the quantity in the limit $L(x,N)$ can be written as $\Phi(x,N)$ In this formula, $\zeta(s)$ is the Riemann zeta function, $\zeta(s,a)$ is the Hurwitz zeta function and $\Psi^{(n)}(s)$ is the $n^{th}$ derivative of the Digamma function. Thanks in advance