A classical problem in number theory is showing that the mean value of an arithmetic function is asymptotic to its mean value over a short interval or over an arithmetic progression, with the interval as short as possible or the modulus as large as possible. We study this problem in the function field setting, and prove for a wide class of arithmetic functions (namely factorization functions), that such an asymptotic result holds, allowing the size of the short interval to be as small as a square‐root of the size of the full interval, and analogously for arithmetic progressions. For instance, our results apply for the indicator function of polynomials with a divisor of given degree, and are much stronger than those known for the analogous function over the integers. As opposed to many previous works, our results apply in the large‐degree limit, where the base field ${\mathbb{F}}_q$ is fixed. Our proofs are based on relationships between certain character sums and symmetric functions, and in particular we use results from symmetric function theory due to Eğecioğlu and Remmel. We also use recent bounds of Bhowmick, Lê and Liu on character sums, which are in the spirit of the Drinfeld–Vlăduţ bound.

中文翻译：

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大程度极限内短区间和算术进展中算术函数的均值

数论中的一个经典问题表明，算术函数的平均值在较短的时间间隔内或在算术级数上渐近于其平均值，且间隔应尽可能短或模数应尽可能大。我们在函数字段设置中研究此问题，并针对一类广泛的算术函数（即因式分解函数）证明了这种渐近结果成立，从而使短间隔的大小可以与函数的平方根一样小。整个间隔的大小，类似地用于算术级数。例如，我们的结果适用于具有给定度数的多项式的多项式的指标函数，并且比针对整数的类似函数的已知指标要强得多。与以前的许多作品不同，我们的结果适用于大程度的限制，其中该基础字段${\mathbb{F}}_q$是固定的。我们的证明基于某些字符和与对称函数之间的关系，尤其是由于Eğecioğlu和Remmel，我们使用了对称函数理论的结果。我们还使用Bhowmick，Lê和Liu的最近界限来表示字符总和，这符合Drinfeld–Vlăduţ界限的精神。