# Generalization of the Riemann zeta function with non-integer bases

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

$$\sum_{n=1}^\infty \frac{1}{{a_n}^s}.$$

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.

## 1 Answer

These series are examples of generalized Dirichlet series, which are series of the form $$\sum a_n \mathrm{e}^{-\lambda_n s},$$ where $a_n, s \in \mathbb{C}$ and $\lambda_n$ is a monotonically increasing sequence of real numbers. Note that this does generalize the Riemann zeta function: with $\lambda_n = \log(n)$ and $a_n = 1$ for all $n$, we obtain $$\mathrm{e}^{-\lambda_n s} = \left(\mathrm{e}^{\log(\frac{1}{n})}\right)^{s} = \frac{1}{n^s} \implies \sum a_n \mathrm{e}^{-\lambda_n s} = \sum \frac{1}{n^s}.$$ There are some convergence results, but (perhaps obviously?) they depend on the coefficients and exponents quite a bit. Typically, it is possible to prove absolute convergence on some right half-plane, then extend analytically to a meromorphic function on some strictly larger domain. Again, the details are going to vary quite a lot.

My recollection is that Apostol treats these in some detail. Lapidus also discusses such series, though I don't recall how much time he actually spends on the details in the text cited below.

• Apostol, Tom M., Modular functions and Dirichlet series in number theory., Graduate Texts in Mathematics, 41. New York etc.: Springer-Verlag. x, 204 p. DM 98.00 (1990). ZBL0697.10023.

• Lapidus, Michel L.; van Frankenhuysen, Machiel, Fractal geometry and number theory. Complex dimensions of fractal strings and zeros of zeta functions, Boston, MA: Birkhäuser (ISBN 0-8176-4098-3/hbk; 978-1-4612-5316-7/pbk). x, 268 p. (2000). ZBL0981.28005.