Let be the Möbius function and , evaluate

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- December 28th 2010, 02:35 PMchiph588@Sum of Möbius Function
Let be the Möbius function and , evaluate

- December 28th 2010, 02:51 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.

- December 29th 2010, 12:26 PMBruno J.
So, is my solution accepted? :)

- December 29th 2010, 12:29 PMchiph588@
- December 29th 2010, 12:37 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.