Mertens function has, by residues, an explicit formula of

$M(n)=\displaystyle\sum_{\rho}\frac{x^\rho}{\rho\zeta'(\rho)}-2+\sum_{n=1}^\infty\frac{(-1)^{2 n}(2\pi)^{2n}}{(2n)! n \zeta(2n+1)x^{2n}}$

where $\rho$ are the zeros of $\zeta(s)$, as usual.

Meanwhile, if we use the following generalized identity for the number of divisors function, $d_z(n)=\displaystyle\prod_{p^\alpha | n}\frac{(z)(z+1)..(z+\alpha-1)}{\alpha!}$ (Aleksander Ivic provides this), it's not much work to see that the Moebius function $\mu(n)$ is equal to $d_{-1}(n)$, and, with $D_z(n) = \sum_{j=1}^n d_z(j)$, that $M(n) = D_{-1}(n)$. Which is to say, we can express $D_{-1}(n)$ in terms of the zeros of the zeta function.

**Here's my question:** is there an explicit formula, for the more general case of $D_z(n)$ for arbitrary z, that the explicit formula for $M(n)$ is a specialization of? Is there some way to express a general explicit formula for $D_z(n)$ in terms of zeta zeroes? Or could there be? Or if not, why not?

Here's motivation for this question:

I want to make a visual, intuitive argument here. Here's a tidy way to express $D_z(n)$ for complex z.

Define

$\displaystyle P_k(n)=\sum_{j=2}^{n}\frac{\Lambda(j)}{\log j} P_{k-1}(\lfloor \frac{n}{j} \rfloor)$ with $P_0(n)=1$

where $\Lambda(j)$ is the Von Mangoldt function. Note by inspection that $P_k(n) = 0$ if $n < 2^k$.

Then our identity for a generalized $D_z(n)$ is

$\displaystyle D_z(n) = \sum_{k=0}^{\lfloor \log_2 n \rfloor}\frac{z^k}{k!}P_k(n)$

(p.28 of P. Bateman and H. Diamond's 2004 edition of “Analytic Number Theory – An Introductory Course” covers this general topic, though not this specific identity).

Now, use this identity to animate, in Mathematica, $\displaystyle\frac{(D_z(n)-1)}{z}$ over the range $z = -1$ to $z = 1$.

**P[n_, k_] := P[n, k] = Sum[ MangoldtLambda[j]/Log[j] P[Floor[n/j], k - 1], {j, 2, n}];P[n_, 0] := 1**

**DD[n_, z_] := Sum[ z^k/k! P[n, k], {k, 0, Log[2, n]}]**

**Animate[DiscretePlot[ (DD[n, z] - 1)/z, {n, 1, 100}], {z, -1, 1, .01}]**

What you'll see, if you watch this animation, is a graph of a function that starts at the function (1-Mertens Function) at z=-1, slowly transforms into the Riemann Prime Counting function right when z=0, and then finally transforms into a straight line, as f(x)=(x-1) at z=1 - all in all, a nice, gradual, fairly smooth transformation between those three important functions.

Now, we have an explicit formula in terms of zeta zeros when z=-1 (due to residues, as given above), and an explicit formula in terms of zeta zeros when z=0 (due to Riemann's original explicit formula for the Riemann Prime Counting function). Given the relatively smooth transition in the animation, it seems very tempting to me to think that there could be an explicit formula using the zeta zeros when $-1<z<0$, connecting these two formulas, or perhaps even more...but I realize it's only an appeal to visuals, of course.

*A Few More Notes*

As visualized above, $\displaystyle \lim_{z \to 0}\frac{D_z(n)-1}{z} = \Pi(n)$, the Riemann Prime Counting Function.

You can show this more easily by taking $d_z(n)=\displaystyle\prod_{p^\alpha | n}\frac{(z)(z+1)..(z+\alpha-1)}{\alpha!}$ and noting that $\displaystyle \lim_{z \to 0}\frac{d_z(n)}{z} = \frac{\Lambda(n)}{\log n}$ except at 1, where the limit is infinity.

There are a bunch of interesting ways to express a generalized $D_z(n)$. I've collected several such ways, with Mathematica demonstrations, on pages 7-11 of this non-rigorous document.