Trigonometric series with arithmetical coefficients
Let \(f\) and \(g\) be arithmetical functions such that \(f(n)=\sum_{d|n}g(d),\forall n\geq 1\). Let \[ B(\theta)=-\sum_{m=1}^\infty\frac{\sin(2\pi m\theta)}{\pi m},\theta\in{\mathbb R}. \] The authors present sufficient conditions for the relation \((D(\theta))\): \[ \sum_{m=1}^\infty\frac{f(m)}{\pi m}\sin(2\pi m\theta)+\sum_{n=1}^\infty\frac{g(n)}{n}B(n\theta)=0. \] For example \((D(\theta))\) holds if \[ \sum_{n=1}^\infty\frac{|g(n)|}{n}<\infty. \] They prove \((D(\theta))\) for \(g=\Lambda\) the von Mangoldt function, \(g=-\mu\log\), with \(\mu\) the Möbius function, and \(g=\tau_z\), with \(\tau_z(n)\) the generic coefficient of the Dirichlet series of \(\zeta(s)^z\), for \(z\in{\mathbb C}\), \(|z+1|\leq 1.\)
scientific article; zbMATH DE number 2112860
scientific article