We investigate the analogues, in \(\mathbb {F}_q[t]\), of highly composite numbers and the maximum order of the divisor function, as studied by Ramanujan. In particular, we determine a family of highly composite polynomials which is not too sparse, and we use it to compute the logarithm of the maximum of the divisor function at every degree up to an error of a constant, which is significantly smaller than in the case of the integers, even assuming the Riemann Hypothesis.

中文翻译：

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$$\pmb {\mathbb {F}}_q[t]$$ F q [ t ] 中的高度复合多项式和除数函数的最大阶数

我们研究了在\(\mathbb {F}_q[t]\) 中的类似物，如 Ramanujan 研究的那样，高度合数和除数函数的最大阶数。特别是，我们确定了一个不太稀疏的高度复合多项式族，我们用它来计算除数函数在每一阶上的最大值的对数，直到一个常数的误差，这明显小于整数的情况，即使假设黎曼假设。