Asymptotic formula and leading constant for the first absolute moment

Determine an asymptotic formula, including a justified leading constant, for the first absolute moment S(X)=∑_{n≤X}|Kl(1,n)| of normalized Kloosterman sums over consecutive moduli.

Background

The main theorem establishes only the order of magnitude S(X)≍X(log X)^{8/(3π)−1}, with unspecified positive comparison constants. The paper then proposes a model constant based on treating Kloosterman factors at distinct prime powers as independent on average.

The authors explicitly state that they do not know a reasonable way to conjecture the asymptotic formula, although they give an audacious model prediction and discuss a related multiplicative mean value that can be analyzed by the Selberg–Delange method. Thus, the precise asymptotic and its leading constant remain unresolved.