Abstract We prove that any q-automatic completely multiplicative function f : N → C essentially coincides with a Dirichlet character. This answers a question of J.-P. Allouche and L. Goldmakher and confirms a conjecture of J. Bell, N. Bruin and M. Coons for completely multiplicative functions. Further, assuming GRH, the methods allow us to replace completely multiplicative functions with multiplicative functions.

中文翻译：

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关于乘法自动序列的注释

摘要 我们证明了任何 q-自动完全乘法函数 f : N → C 本质上与 Dirichlet 字符重合。这回答了 J.-P. Allouche 和 L. Goldmakher 并证实了 J. Bell、N. Bruin 和 M. Coons 对完全乘法函数的猜想。此外，假设 GRH，这些方法允许我们用乘法函数替换完全乘法函数。