Let $\kappa:\lbrace 1,2,3,\ldots\rbrace\longrightarrow \mathbb Z$
be the convolutional inverse in the Dirichlet ring of $n\longmapsto {n+1\choose 2}$. It is defined by $\kappa(1)=1$ and
by the functional equation $\sum\_{d\vert n}{d+1\choose 2}\kappa(n/d)=0$ for $n\geq 2$.


It is not hard to show that $\kappa(3\cdot 5\cdot p^2)=0$
for every prime number $p\geq 7$.


There seem to be very few other values $n$ such that $\kappa(n)=0$.
My computer found only two : $2^4\cdot 13\cdot 23$ and $5^3\cdot 7\cdot 37$.


*Is the set of such exceptional zeros of $\kappa$ finite or not?*



Exceptional zeros of a convolutional inverse
There are many more examples. An infinite series is for $n=5^8\cdot7\cdot p^2$ for primes $p$ different from $5$ and $7$.


Another infinite series is $n=5\cdot11\cdot17^3\cdot p^2$ for primes $p$ different from $5,11,17$. And there are many more infinite series.


