Where can I find papers and research on the Alternating Hurwitz Zeta Function?

Question

The function is as follows (I don't know it's name, it could be 'Generalised Dirichlet Eta Function') $$f(s,q)=\sum_{n=0}^{\infty} \frac{(-1)^{n}}{(n+q)^{s}}$$

Answer 1

Where can I find papers and research on the Alternating Hurwitz Zeta Function?

Probably, a good starting point is at MathSciNet and ScienceDirect, which give, for example:

---

• Identities for the Hurwitz zeta function, Gamma function, and L-functions

---

• Another discrete Fourier transform pairs associated with the Lipschitz–Lerch zeta function

---

Answer 2

As noted in the comments, your function is related to the Lerch transcendent $\Phi (z,s,q)$. Here $$\Phi(z,s,q) = \sum_{n = 0}^\infty \frac{z^n}{(n + q)^s}.$$ Setting $z = -1$ gives $$\Phi (-1,s,q) = \sum_{n = 0}^\infty \frac{(-1)^n}{(n + q)^s},$$ which is your $f (s,q)$.

Much is known about the Lerch transcendent. For example, when $q = 1$ we have $$\Phi(-1,s,1) = \eta (s),$$ where $\eta (s)$ is the Dirichlet eta function.