This paper provides a mean value theorem for arithmetic functions f defined by f(n) =∏d|ng(d), where g is an arithmetic function taking values in (0, 1] and satisfying some generic conditions. As an application of our main result, we prove that the density μ(n) (respectively, ρq(n)) of normal (respectively, primitive) elements in the finite field extension 𝔽qn of 𝔽q are arithmetic functions of (nonzero) mean values.

中文翻译：

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一类密度类算术函数的均值定理

本文提供了算术函数的中值定理F被定义为 F(n) =∏d|nG(d), 在哪里G是一个算术函数，取值(0, 1]并满足一些一般条件。作为我们主要结果的应用，我们证明了密度μ(n)（分别，ρq(n)) 有限域扩展中的正常（分别为原始）元素𝔽qn的𝔽q是（非零）平均值的算术函数。