$$\lim_{x\rightarrow+\infty}\zeta(-a,x-a)+\zeta(-a,x+a)-2~\zeta(-a,x)=0$$ I was trying to solve an infinite summation problem of quadratic radicals, in which I used the Hurwitz Zeta Function to try to solve the problem, but in which I inevitably ran into this limit. Of course, this is true in that case if only $a=\frac12$ is needed, but how do you generalize it to $a\in(0,1)$?
My idea is to do an analytic extension of this Hurwitz Zeta Function, and then expand the result of the extension to try to solve it. But I found that if you do that, then it loses its symmetry and its simple form, and becomes a more complicated problem, at least I think, going back to the problem of quadratic radical summation. Of course, this may just be my understanding.
Can anyone solve this problem about the limits of Hurwitz Zeta Function? Thank you very much!