An exponential sum involving primes

Suppose

$$\sum_{p<X} e(2\sqrt{p}) \ll X^\alpha .$$

If that bound held for some $\alpha<\frac34$ then one could extend the current results on the 1-level density of low-lying zeros of cusp form $L$-functions. This would imply (assuming GRH for the critical zeros of these $L$-functions) that $L(f,s)$ has no Siegel zeros in a strong sense. Namely, that there exists $\delta>0$ such that $L(f,s)$ is nonzero for $s>1-\delta$.

(References and more details on the above are sought).

It is tempting to think that the above estimate must surely hold for any $\alpha>\frac12$, 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 $\lambda_f(n)$ denote the (suitably normalized) Fourier coefficients of a holomorphic cusp form. Then [Conrey and Ghosh?, unpublished?] have shown that

$$\sum_{p<X} \log p \lambda_f(p) e(2\sqrt{p}) \sim c X^{3/4} .$$

The lack of cancellation in that sum is puzzling.

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