Suppose that $a_n$ is a strict monotonic increasing sequence of real numbers, with the index starting at $n=1$. Then we can look at the infinite series
For the sequence $a_n = n$, this corresponds to the Riemann zeta function, and for some constant $q$, the sequence $a_n = n+q$ is the Hurwitz zeta function. However, this function is more general.
Do these "deformed zeta functions" have a name, and have they been studied? Is it known when they do and don't converge?
For reference, these emerge quite naturally in music theory. The Riemann zeta function has an interesting practical use in that it can be interpreted as a plot of how well the various equal divisions of the octave approximate the harmonic series. The generalization presented here is useful in telling us how well they approximate deformed harmonic series, or in musical terms, inharmonic timbres such as a bell, marimba, etc.