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On Ramanujan sums of a real variable and a new Ramanujan expansion for the divisor function
The Ramanujan Journal ( IF 0.6 ) Pub Date : 2021-05-06 , DOI: 10.1007/s11139-021-00412-z Matthew S. Fox , Chaitanya Karamchedu
中文翻译:
在Ramanujan上,一个除数函数的实变量和一个新的Ramanujan展开的和
更新日期:2021-05-06
The Ramanujan Journal ( IF 0.6 ) Pub Date : 2021-05-06 , DOI: 10.1007/s11139-021-00412-z Matthew S. Fox , Chaitanya Karamchedu
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:
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。