What's about the generating function for the Mertens function?

I don't know if it's possible to deduce an answer to this question. Feel free to refer literature or to argue if it makes sense.

Wikipedia page for Hamonic number tell us that what are the generating functions for harmonic numbers, this entry provide us two of such generating functions (ordinary and exponential) that you can read and study in the corresponding section $3$.

On the other hand, let $\mu(n)$ the Möbius function, and as you read in this MarthWorld, the Mertens function is defined by $$M(n)=\sum_{1\leq k\leq n}\mu(k).$$

Question. Do you know what's about a generating function for the Mertens function?

I am interested also if makes sense a generating function for $\sum_{k=1}^n\frac{\mu(k)}{k}$, thus feel free to add your thoughts as comments also for this second question. Many thanks.

• There are typically not generating functions (OGF, EGF) in the combinatorial sense for most number theoretic functions, nor summatory functions of number theoretic functions. You can, however, recover these summatory functions via an inverse Mellin transformation. See the Wikipedia page for the Mertens function for an example which expresses $M(x)$ exactly as a sum over the non-trivial zeros of the Riemann zeta function. A similar argument should give you the formula for $\sum_{n \leq x} \mu(n) / n^t$ by the Residue theorem since $\sum_{n \geq 1} \mu(n) / n^{s+t} = 1 / \zeta(s+t)$. – mds Jun 2 at 10:56