Is the integral $\int_1^{\infty} {A(t)}{t^{-s-1}}dt$ a holomorphic function of $s$?

Asked Jan 13 '15 at 13:10

Active Jan 13 '15 at 13:46

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My question is whether, for $Re(s)=\sigma > 3/4$, $$s\int_1^{\infty} \dfrac{A(t)}{t^{s+1}}dt$$ is holomorphic, where $A(x)=O(x^{3/4})$.

Under absolute value, it is easy to see that the integral converges to some real number. But I am not sure if without it.

edited Jan 13 '15 at 13:46

asked Jan 13 '15 at 13:10

Tonoro

Try to use Morena's theorem. See here. – Mhenni Benghorbal Jan 13 '15 at 13:33
Check this. – Mhenni Benghorbal Jan 13 '15 at 13:38
According to that page, my $f(z)$ is $\int_1^{\infty} \dfrac{A(t)}{t^{z+1}}dt$? – Tonoro Jan 13 '15 at 15:21

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