A long time ago now, I wondered whether or not there exists some sequence of real numbers $(a_n)_{n \in \mathbb{N}}$, different from the zero sequence, such that for any $m \in \mathbb{N}$, $$ \sum_{n=1}^{\infty} \, a_{nm} = 0. $$ To avoid any confusion: I really mean the product $nm$ in the subscript, it does not indicate an array of numbers. A friend of mine found a solution to this problem:
$a_n = \mu(n)/n$, where $\mu$ denotes the Möbius function, should work.
Now, this makes it easy to construct a function $f : \mathbb{R} \to \mathbb{R}$, different from the zero function, with the property that $$ \sum_{n=1}^{\infty} f(\alpha n) = 0, $$ for all $\alpha \in \mathbb{R}$, but it would of course by no means be continuous. Indeed, we may set $f(n) = a_n$ for all $n \in \mathbb{N}$ and extend by zero everywhere else.
Therefore my question is: does such a continuous function exist?