Let be the Möbius function and , evaluate

Printable View

- Dec 28th 2010, 03:35 PMchiph588@Sum of Möbius Function
Let be the Möbius function and , evaluate

- Dec 28th 2010, 03:51 PMBruno J.
We can factor this as where means that but . Clearly the product is unless for each . Hence the sum is if is not divisible by a -power (other than ), and otherwise.

- Dec 29th 2010, 01:26 PMBruno J.
So, is my solution accepted? :)

- Dec 29th 2010, 01:29 PMchiph588@
- Dec 29th 2010, 01:37 PMBruno J.
Well it's more that I'm assuming the reader will have no trouble convincing him/herself of it! (Happy)

Each term in the sum is , where and is such that . In the product I wrote, the term will be obtained as once the product is expanded. Conversely, each term in the expansion of the product can be found in the sum.