(References and more details on the above are sought).
It is tempting to think that the above estimate must surely
hold for any
, because one naturally expects
square-root cancellation in any ``random looking'' sum.
However, the following example shows that nature is more subtle
than that. Let denote the (suitably normalized)
Fourier coefficients of a holomorphic cusp form.
Then [Conrey and Ghosh?, unpublished?] have shown that
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