I am posting the following question from Math.Stackexchange:

Let $f$ be a $1$-periodic function, i.e., $f(x+1)=f(x)$, defined on the interval $(0, 1)$ by the formula $$ f(x)=2x-1. $$ For a real number $x$ consider the series $$ \sum_{n=1}^\infty\frac{f(nx)}{n}. $$ It is easily seen that the series converges if $x$ is rational and if $f$ is assumed to vanish at all integers.My question is:

does the convergence hold for irrational$x$?I would also be grateful for references to any results about kindred objects.

I am also interested in a description of the set of all $x$ for which the series converges, and in properties of the function determined by the series.