An Abel-Tauber theorem on convolutions with the Möbius function

Abstract

Suppose n : R + → R + n:{R^ + } \to {R^ + } and n ( x ) / x n(x)/x is integrable on ( 0 , ∞ ) (0,\infty ) . For s > 0 s > 0 we define \[ n ~ ( s ) = s ∫ 0 ∞ e − u s 1 − e − u s n ( u ) d u . \tilde n(s) = s\int _0^\infty {\frac {{{e^{ - us}}}}{{1 - {e^{ - us}}}}} n(u)du. \] In this paper an Abel-Tauber theorem is proved concerning this transform. Moreover the relation between n ~ ( s ) \tilde n(s) and Σ m ⩽ s n ( s / m ) / m {\Sigma _{m \leqslant s}}n(s/m)/m is studied.