I would like to prove uniform convergence of function series like :


$$\sum\limits\_{n=1}^{\infty} \chi(n) f(nx)$$ where $\chi$ is a primitive character and $f(x)$ a function decreasing to zero in zero and infinity.


Typically: $$F(x)= \sum\limits\_{n=1}^{\infty} \chi(n) n^2 x^2 e^{-nx}$$


On intervals $[\epsilon, \infty[$ with $\epsilon >0$ it is obvious that we have uniform convergence as for high $P$ we have a constant $M$, such that $\forall x>\epsilon$: $$|\sum\limits\_{n=P}^{\infty} \chi(n) n^2 x^2 e^{-nx}| \le M \sum\limits\_{k=P}^{\infty} n^2 \epsilon^2 e^{-n \epsilon }$$


But on $[0,\epsilon]$ it is less obvious... Thanks to twisted Poisson Summation formula applied with Dirichlet Character we know that $F(x)$ is well defined in zero with $\lim\_{x \to 0} F(x) =0$ but what about uniform convergence ? (For info about twisted Poisson Summation formula [Extension of Poisson Summation formula](https://mathoverflow.net/questions/144563/extension-of-poisson-summation-formula))


I tried to use Mellin transform to find a proof but I did not succesd yet. Any idea or reference on this subject? 



Uniform convergence of infinite sum with Dirichlet characters
Your specific series is not uniformly convergent on any interval of the form $[0,\delta]$. If it were, then it would be uniformly Cauchy on that interval; in particular, there would exist $P$ such that for all $M\ge N>P$ we would have
$$
\bigg| \sum\_{n=N}^M \chi(n)n^2x^2e^{-nx} \bigg| < \frac12
$$
for all $x\in [0,\delta]$. However, we can take $M=N$ and $x=\frac1N$, yielding the single term $|\chi(N)e^{-1}|$ which is too large (for the infinitely many $N$ at which $\chi$ does not vanish).


This counterexample seems to generalize to any $f(nx)$ not identically zero, and uses very little about $\chi$ other than that $\limsup|\chi(n)|>0$.


