Power series with the von Mangoldt function

Abstract

We study the analytic behavior of a power series with coefficients containing the von Mangoldt function. In particular, we extend an explicit formula of Hardy and Littlewood for related functions and derive further representation formulas in the unit disk that reveal logarithmic singularities on a dense subset of the unit circle. As an essential tool for proving the square integrability of occurring limit functions together with respective error estimates we contribute a new proof of a Ramanujan-like expansion of an arithmetic function consisting of the von Mangoldt function and the Euler function.