# An exponential sum involving primes

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 Then [Conrey and Ghosh?, unpublished?]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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