On some Diophantine Fourier series (Q606279)

Let \[ \overline {B}_l(x)=\sum_{k=0}^l\binom{l}{k}B_{l-k}\{x\}^k, \] with \(B_ {l-k}\) the \((l-k)\)-th Bernoulli number and \(\{x\}\) the fractional part of \(x\). The authors prove that \[ \sum_{n=1}^\infty\frac{\mu(n)}{n^z}\overline {B}_l(nx)=-\frac{l!}{(2\pi i)^l}\sum_{n=-\infty\atop n\neq 0}^\infty\frac{\sigma_{l-z}^*(n)}{n^l}e^{2\pi i nx}, \] for every \(0<c<1\), \(\text{Re}(z)>1-c\), where \(\sigma_{l-z}^*(n)=\sum_{d|n}\mu(d)d^{l-z}\), \(\mu(n)\) the Möbius function.