I know that a similar question was in Riemann's thesis (see if you have the paper [$1$] where it is referred) Could you provide details of this new example in my Question?
Edited: I don't know how define $ \left\{ x \right\} $, see Riemann thesis and the first section of [1], because is defined a function $ \left\{ x \right\} $, to get a similar example.
Question. How do you calculate the points where $$f(x)=\sum_{n=1}^\infty\frac{\left\{n x \right\}}{n^4}$$ is discontinuous? And, what is the jump in each of these points? Thanks.
Thus I am asking an easy way to get a closed form of the points of discontinuity of our $f(x)$ and how one can calculate the jusp in each of them.
References:
[$1$] Córdoba, Encounters at the interface between Number Theory and Harmonic Analysis, Proceedings of the "Segundas Jornadas de Teoría de Numeros" (Madrid, 2007), Biblioteca de la Revista Matematica Iberoamericana. I say the page before the second section.
Let's observe the jump at a rational point $x=p/q$, where $\gcd(p,q)=1$. The function $\{x\}$ has a jump of $-1$ at every whole number. So, for $x=p/q$, the jumps are at $n=mq$ for each $m$. Renumber the sum and you get:
$$\text{jump}(p/q)=\sum_{m=1}^\infty \frac{-1}{(mq)^4}=-\frac{1}{q^4}\zeta(4)$$