Prime numbers in two bases
For two coprime integer bases \(q_1, q_2 \ge 2\), let \(f\) be a proper strongly \(q_1\)-multiplicative function and \(g\) a strongly \(q_2\)-multiplicative function, for example exponentiated sum-of-digits functions in bases \(q_1\) and \(q_2\). The main result states that \[ \Big|\sum_{n\le x} \Lambda(n) f(n) g(n) \exp(2\pi i\vartheta n)\Big| \ll x \exp(-c \log x/\log \log x) \] uniformly for \(\vartheta \in \mathbb{R}\), where \(\Lambda\) denotes the von Mangoldt function. Here, the positive constant \(c\) and the implicit constant depend on \(f\) and \(g\). The same result holds for the Möbius function instead of the von Mangoldt function. In particular, \(f(n)g(n)\) is orthogonal to the Möbius function. Since these sequences are produced by zero-entropy dynamical systems, they provide a new class satisfying the Sarnak conjecture. For certain multiplicative functions, more explicit upper bounds and a prime number theorem are given. The proofs combine Fourier analysis, Diophantine approximation, and combinatorial arguments.
scientific article; zbMATH DE number 7226651
scientific article