Let \(q\geq 2\) be a fixed integer. Any integer \(n\) can be represented uniquely in the negative base \(-q\), i.e. in the form \(n=c_0+c_1(-q)+\cdots +c_h(-q)^h\), where \(c_i\in \{0,1,\dots ,q-1\}\). The authors investigate the sum of digits function \(\nu_{- q}(n)=c_0+c_1+\cdots +c_h\). They show, among other things, that \[ \sum_{n<N} (\nu_{-q}(n)-\nu_{-q}(-n)) =NG(\log_{q^2} N) +O(\log^2 N), \] where \(G(x)\) is a continuous, periodic, nowhere differentiable function, and obtain a Gaussian asymptotic distribution result for \(\nu_{-q}(n)-\nu_{-q}(-n)\).
The proofs use automata and analytic methods.