The Furstenberg correspondence in this case is basically saying that you have something like a Fourier transform for multiplicative functions — different measure, and instead of R or C, we have Z^+ — but you can still say things like “The Fourier transform of a convolution is the product of the Fourier transforms of the components”.

You then use a “basic” argument to show that the odd moments of the Fourier transform of \lambda(n) have to vanish by symmetry. That’s the stuff regarding f_0, I think. Then you have to show that throwing in arbitrary phases into the moment calculation doesn’t throw things off — is that the minor arc?

I’m trying to figure out how to translate your even-length correlation results into these terms.

Please let me know if this is even remotely accurate.

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