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Consider the Dirichlet series (for fixed $0 < a \leq 1$):

$$\zeta_a(s) = \sum_{n\geq 1}\frac{a^n}{n^s}$$

which reduces to the Riemann zeta function for $a=1$. What is known about this function, in terms of functional equations or relations to other standard Dirichlet series? If we try to do the standard trick with Mellin transforms to relate this to a Jacobi theta function, it works and we get that: $$\pi^{-s/2}\Gamma(s/2)\zeta_a(s) = \int_0^\infty\frac{\nu(z,iy)-1}{2}y^{s/2}\frac{dy}{y}$$ for $\nu(z,\tau)$ the Jacobi theta function and $\exp(2\pi iz) = a$: $$\nu(z,\tau) = \sum_{n\in \mathbb Z}\exp(\pi in^2\tau + 2\pi inz).$$ Is there some nice functional equation with non zero $z$ for $\nu(z,\tau)$?

The motivation for this post comes from this blog post of Matt baker where he shows that for $a$ integral, $\sum_{d|n}\mu(n/d)a^d \equiv 0 \pmod n$ as a generalization of Fermat and Euler's theorem.

This suggests that the arithmetic function $n \to a^n$ has interesting arithmetical properties and perhaps the corresponding Dirichlet function would be interesting. Unfortunately, $\zeta_a(s)$ is divergent for $a > 1$, the regime we actually care about!

So perhaps, we should instead look for p-adic analogues. More precisely, for $0 < a < 1$ a rational number such that $a-1$ is a p-adic unit. I believe I can show in this case that for $n$ a negative integer, $\zeta_a(n)$ takes rational values that are moreover p-adically integral and that there is a p-adic analytic function that interpolates these values.

Have these functions been studied before?

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    $\begingroup$ This is a version of the Lerch zeta function, see en.wikipedia.org/wiki/Lerch_zeta_function $\endgroup$
    – Stopple
    May 27, 2020 at 13:54
  • $\begingroup$ In the space of (degree $1$..) Dirichlet series with functional equation we can take linear combinations so it is continuous, but when we add the Euler product requirement it becomes a discrete space. Also the RH has a formulation which holds for linear combination of L-functions, replacing the convergence/asymptotic of the Euler product by the convergence/asymptotic of $\sum_{n,\gcd(n,k!)=1} a_nn^{-s}$ as $k\to \infty$ (so it doesn't correspond to the zeros of $\sum_n a_nn^{-s}$ but to the zeros of each L-function in the linear combination) $\endgroup$
    – reuns
    May 27, 2020 at 22:13

1 Answer 1

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This is the Polylogarithm, valid for arbitrary complex order $s$ and for all complex arguments $a$ with $|a| < 1$; it can be extended to $|a| \ge 1$ by the process of analytic continuation. It is related to the Lerch zeta function. The functional equation is originally due Jonquiere in 1889. The fullest investigation of the functional equation I can find is in this talk by Lagarias. There is also this monograph by Laurincikas.

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  • $\begingroup$ The polylogarithm is $\zeta_a(s)$ as a function of $a$, for fixed $s\in\mathbb{N}$. $\endgroup$ May 27, 2020 at 20:35

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