We show that the absolute convergence of a Ramanujan expansion does not guarantee the convergence of its real variable generalization, which is obtained by replacing the integer argument in the Ramanujan sums with a real number. We also construct a new Ramanujan expansion for the divisor function. While our expansion is amenable to a continuous and absolutely convergent real variable generalization, it only interpolates the divisor function locally on \({\mathbb {R}}\).

中文翻译：

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在Ramanujan上，一个除数函数的实变量和一个新的Ramanujan展开的和

我们表明，Ramanujan展开的绝对收敛不能保证其实变量泛化的收敛，这是通过将Ramanujan sum中的整数参数替换为实数而获得的。我们还为除数函数构造了一个新的Ramanujan扩展。尽管我们的扩展适合于连续且绝对收敛的实变量泛化，但它仅在\（{\ mathbb {R}} \）上局部插入除数函数。