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Is there some simplifying expression for the sum

$$\sum_{n=1}^\infty \frac{1}{|an+b|^x}$$

where $a,b$ are arbitrary real numbers, and $x$ is a real number larger than 1 (which should ensure convergence)? (This is a more general version of the question Calculating squared reciprocals of arithmetic series , and I have not found it brought up in other threads.) Going one step further, is there an expression for the sum

$$\sum_{n=1}^\infty \frac{n}{|an+b|^y},$$

as well, where $y$ is a real number larger than 2? I see that in principle one could approximate the sums by integrals, but are there any exact expressions in terms of more "well-known" (even if not necessarily elementary) functions?

helloworld
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    It is the Hurwitz zeta function, no simplification, but an analytic continuation and functional equation. It is clear how to relate the second series at $y=x$ to the first series at $x$ and $x-1$. – reuns Apr 25 '22 at 17:05
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    Try taking the derivative with respect to $a$ of the first function and see how that relates to the second function – Henry Apr 25 '22 at 17:13
  • Thanks! That's a good point; I'd tried the differentiation trick but then set it aside because I was working in a context where $x$ and $y$ do not differ by $1$...somehow I entirely overlooked the fact that it still simplifies the second summation in terms of $y-1$, which is good enough. – helloworld Apr 25 '22 at 22:28

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