The zeta-dimension of a set A of positive integers is Dim _ζ (A) = inf {s|ζA(s) < ∞}, where ζA(s)= ∑_(n∈A) n~(-s). Zeta-dimension serves as a fractal dimension on Z~+ that extends naturally and usefully to discrete lattices such as Z~d, where d is a positive integer. This paper reviews the origins of zeta-dimension (which date to the eighteenth and nineteenth centuries) and develops its basic theory, with particular attention to its relationship with algorithmic information theory. New results presented include a gale characterization of zeta-dimension and a theorem on the zeta-dimensions of pointwise sums and products of sets of positive integers.
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机译：SET A A的正整数的Zeta维度为Dim_∈（a）= Inf {s |ζA（s）<∞}，其中ζa（s）=σ_（n∈a）n〜（-s）。 Zeta-尺寸用作Z〜+上的分形尺寸，其自然而然地延伸到离散的格子如Z〜D，其中D是正整数。本文审查了Zeta-Dimension（第十八和第十几个世纪的日期）的起源，并开展了其基本理论，特别关注其与算法信息理论的关系。呈现的新结果包括Zeta-尺寸的大风特征和Zeta-尺寸的Zeta-尺寸和正整数集合的产品的定理。
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