Let $N$ be a Poisson random variable with rate $\lambda >0$. Set $X:=N\pmod m$. What is the pmf and expectation of $X$?
For $j=0,1,\dotsc,m-1$, we have $\{X=j\}=\bigcup_{k=0}^\infty \{N=j+mk\},$ so $$P(X=j)=\sum_{k=0}^\infty P(N=j+mk)=e^{-\lambda} \sum_{k=0}^\infty \frac{\lambda^{j+mk}}{(j+mk)!},$$ which converges but I can't find its sum. For $\lambda^m<1$, the estimates, $$P(X=j)\leq \frac{1}{1-\lambda^m} P(N=j)$$ and $$\mathbb{E}(X)\leq \frac{\lambda}{1-\lambda^m},$$ hold. These were derived by considering the geometric series for $1/(1-\lambda^m)$ and comparing the two... but this doesn't help with the original problem...
Is there anyway to continue or an alternate route of attack? Resources for studying discrete random variables modulo $m$ in general?