Monotonicity properties of ‐functions

Abstract

We present some monotonicity results for a class of Dirichlet series generalizing previously known results. The fact that ζ′false(sfalse) is in that class presents a first example of an arithmetic function for which the associated Dirichlet series is completely monotonic, but not logarithmically completely monotonic. Lastly, we use similar techniques to prove another formulation of the Riemann hypothesis for the L‐function associated to the Ramanujan‐tau function.