Suppose

If that bound held for some then one could extend the current results on the 1-level density of low-lying zeros of cusp form -functions. This would imply (assuming GRH for the critical zeros of these -functions) that has no Siegel zeros in a strong sense. Namely, that there exists such that is nonzero for .

(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Then [Conrey and Ghosh?, unpublished?]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

The lack of cancellation in that sum is puzzling.

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