We introduce a mean counting function for Dirichlet series, which plays the same role in the function theory of Hardy spaces of Dirichlet series as the Nevanlinna counting function does in the classical theory. The existence of the mean counting function is related to Jessen and Tornehave's resolution of the Lagrange mean motion problem. We use the mean counting function to describe all compact composition operators with Dirichlet series symbols on the Hardy–Hilbert space of Dirichlet series, thus resolving a problem which has been open since the bounded composition operators were described by Gordon and Hedenmalm. The main result is that such a composition operator is compact if and only if the mean counting function of its symbol satisfies a decay condition at the boundary of a half-plane.

我们介绍了Dirichlet级数的均值计数函数，该函数在Dirichlet级数的Hardy空间的函数理论中的作用与经典理论中的Nevanlinna计数函数相同。均值计数函数的存在与Jessen和Tornehave对拉格朗日均运动问题的解决有关。我们使用均值计数函数在Dirichlet级数的Hardy-Hilbert空间上用Dirichlet级数符号描述所有紧致合成算子，从而解决了自Gordon和Hedenmalm描述有界合成算子以来一直存在的问题。主要结果是，当且仅当其符号的平均计数函数满足半平面边界处的衰减条件时，这种合成算子才是紧凑的。