We introduce various Ruelle type zeta functions ζL (s) according to a choice of homogeneous "length functions" for a lattice L in ℂ via Euler products. The logarithm of each ζL (s) yields naturally a certain arithmetic function. We study the asymptotic distribution of averages of such arithmetic functions. Asymptotic behavior of the zeta functions at the origin s = 0 are also investigated.
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