论文标题
关于Ramanujan的真实变量和新的Ramanujan扩展的总和
On Ramanujan Sums of a Real Variable and a New Ramanujan Expansion for the Divisor Function
论文作者
论文摘要
我们表明，Ramanujan扩展的绝对收敛并不能保证其实际变量概括的收敛性，这是通过用实际数字替换Ramanujan中的整数参数来获得的。我们还为除数函数构建了新的Ramanujan扩展。尽管我们的扩展适合连续且绝对收敛的真实变量概括，但它仅在$ \ mathbb {r} $上本地插入了除数函数。
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 $\mathbb{R}$.
