Given two integers a and k > 0, the number of factorizations of a (mod k) is the number of ordered pairs (s, t) ∈ {0, 1, . . . , k − 1}² satisfying s · t ≡ a (mod k). This number is known to be expressible by a formula involving the greatest common divisor function. Motivated by such a formula, we derive several formulae counting the number of factorizations of a (mod k) subject to certain other natural restrictions. Some of these formulae are obtained as consequences of finite Fourier series expansions of the greatest common divisor function, whereas some are shown to be closely connected with the notion of unitary divisors.

中文翻译：

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整数分解和有限傅立叶级数展开

给定两个整数一个和K> 0，的因式分解的数目一（MOD ķ）是有序对的数量（S，T）∈{0 ， 1 ，。。。，k − 1} ²满足s·t≡a（mod k）。已知该数字可由包含最大公约数函数的公式表示。受这样一个公式的激励，我们导出了几个公式，计算一个（mod k），但受某些其他自然限制。这些公式中的一些是最大公因数函数的有限傅里叶级数展开的结果，而另一些则被证明与unit除数的概念紧密相关。