Let
where is Liouville's function.
How do I evaluate the sum
Thanks in advance.
Here's a broad hint: First, do you know the identity
? It may be found in the Wikipedia article on Liouville's function, or as Theorem 300 in Hardy and Wright.
Next, write your sum as
Write out the first nine or so terms and collect together terms with , , and so on. Simplify, using the identity
where is the Riemann zeta function. You should end up with an expression that involves the identity that I mentioned at the beginning of this post (with s = 2). Finally, use the well-known identities
and
to arrive at the final answer.
An Introduction to the Theory of Numbers by Hardy and Wright.
Thanks for your hint, Petek.
I'm not sure I follow
When I write out the terms of the sum, I can see that this is equal to , but can't see how to evaluate it from here.You should end up with an expression that involves the identity that I mentioned at the beginning of this post (with s = 2).
Am I right in thinking that
so that when s=2
Regards