Sign-changes of the Thue-Morse fractal function and Dirichlet \(L\)-series
Denoting by \(s(n)\) the sum of the binary digits of \(n\), the intriguing sequence \((- 1)^{s(3n)}\) has been studied by Newman, Newman and Slater, and Coquet who successively proved that \(\sum_{0 \leq n \leq N - 1} (- 1)^{s(3n)}\) is strictly positive, then of the order of \(N^{\log 3 \over \log 4}\) and finally that \[ \sum_{0 \leq n \leq N - 1} (- 1)^{s(3n)} = N^{{\log 3 \over \log 4}} \psi \left( {\log N \over \log 4} \right) + O(1) \] where \(\psi\) is a function of period 1, continuous and nowhere differentiable. Furthermore \(\psi\) is strictly positive [see \textit{J. Coquet}, Invent. Math. 73, 107-115 (1983; Zbl 0528.10006)]. Many works have been devoted to trying to generalize these statements to sums \(\sum_{0 \leq n \leq N - 1} (-1)^{s(pn + b)}\), where \(p\) is a prime number. In particular an expression analogous to Coquet's has been obtained for \(b=0\) and if 2 is a primitive root modulo \(p\) by \textit{S. Goldstein}, \textit{K. A. Kelly} and \textit{E. R. Speer} [J. Number Theory 42, 1-19 (1992; Zbl 0788.11010)]. The authors of the paper under review prove that if 2 is a primitive root modulo \(p\), with \(p \neq 3,5\), or if 2 generates all squares modulo \(p\), with \(p \neq 17, 41\) then \(\sum_{0 \leq n \leq N - 1} (-1)^{s(pn)}\) takes both positive and negative values. Furthermore they give fine results for the zeroes of the analogues of Coquet's \(\psi\), study briefly sums \(\sum (-1)^{s(pn + i}\) and the twisted sum \(\sum(-1)^{s(n)} {n + 1 \choose p}\), where \({n+1\choose p}\) is the Legendre symbol.
scientific article; zbMATH DE number 784056
scientific article