I am wondering if I could deduce the bound for the partial sums \[ \sum_{n \leq x}a(n) \ll x^{A}, \quad x \to \infty \] from the relation \[ \sum_{n \geq 1}a(n)e^{-ny} \ll y^{-A}, \quad y \to 0^{+}. \]

The condition on $a(n)$ is $|a(n)| = 1$ for all $n \geq 1$.

If anything has been already established on this, it could save my time tremendously (and much more, might make me happy).

You could use the fact that the function \[ \sum_{n \geq 1}\frac{a(n)}{n^{s}} \] is analytic for $\sigma > A$.

It is a routine that via Mellin inversion theorem, \[ \sum_{n \geq 1}a(n)e^{-ny} = \frac{1}{2 \pi i }\int_{(3)} \Gamma(s) f(s) y^{-s}ds. \]

My first question should be, is it ever possible?

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