@robjohn The only 'slightly' related thing I can say I have done in the past was I used hurrwits fourier expansion for bernoulli polynomials
$$\text{B}'_{2r}(x)=\frac{-r!}{(2\pi i)^r}\sum_{k=-\infty}^\infty'\frac{e^{2\pi i x}}{k^r}$$
And properties of dirichlet convolutions one can show identitys for arithmetic functions like,
$$\frac{(-1)^{r+1}2(2r+1)!}{(2\pi)^{2r+1}}\sum_{n=1}^\infty\frac{(f*1)(n)\sin(2 \pi n x)}{n^{2r+1}}=\sum_{n=1}^\infty\frac{f(n)}{n^{2r+1}}\text{B}'_{2r+1}(nx)$$
$$\frac{(-1)^{r+1}2(2r)!}{(2\pi)^{2r}}\sum_{n=1}^\infty\frac{(f*1)(n)\cos(2 \pi n x)}{n^{2r}}=\sum_{n=1}^\i…