Do Riemann-Weil formulas exist for functions other than the Mangoldt function $ \Lambda (n) $

Question

Are there formulas similar to the Riemann-Weil formula for other arithmetical functions like $ \mu (n) $ or $ \lambda (n) $, for example a sum of the form $ \sum_{n=1}^{\infty}a(n) f(n) $ with this sum being related to the sum over the imaginary part of the Riemann zeros involving the Fourier transform of $ f(x)$?

To summarize: can the Riemann-Weil explicit formulas be generalized to other arithmetical functions? For example:

$$ \sum_{n=1}^{\infty} \frac{\mu(n)}{\sqrt{n}}g(\log n)=\sum_{\gamma}\frac{h( \gamma)}{\zeta '( \rho )}+\sum_{n=1}^{\infty} \frac{1}{\zeta ' (-2n)} \int_{-\infty}^{\infty}dxg(x)e^{-(2n+1/2)x} $$

Answer 1

The reason that $\mu(n)$ and $\lambda(n)$ have such expressions is that the corresponding Dirichlet generating functions can be expressed in terms of the Riemann zeta function: $$ \sum_n \frac{\mu(n)}{n^s}=\frac{1}{\zeta(s)} $$ and $$ \sum_n \frac{\lambda(n)}{n^s}=\frac{\zeta(2s)}{\zeta(s)}. $$ In general, if the Dirichlet series coefficients are multiplicative: $a(mn)=a(m)a(n)$ for $(m,n)=1$, then there will be an Explicit Formula in terms of the zeros (and poles) of the corresponding $L$-function. For example, when $a(n)=\chi(n)$ is a Dirichlet character modulo $d$, there is a formula involving the zeros of $L(s,\chi)$. The classical version (with a cutoff for the sums rather than a test function and its transform) is in Montgomery and Vaughan's Multiplicative Number Theory, chapter 12.1. You can imitate the treatment of the zeta function in chapter 12.2 to get a version with a test function. See also the notes in chapter 12.3, which point to Weil's treatment of the case of an $L$-function attached to a Grossencharaktere $\chi$.