In the study of the Riemann zeta function (the connection here is not really visible), I encountered in Titchmarsh book about the Riemann zeta function the integral
$$ \int_{N}^{\infty} \frac{\sin y}{y^{s+1}} dy. $$ If i use integral by parts the power decrease but how can I obtain the integral.
In math wolfram it say is related to the incomplete gamma function.
I think an approach could be $sin(z)= e^{iz}-e^{-iz}/2i$ that came similar to gamma function but with complex values for e.
This is effectively related to the incomplete gamma function.
Consider that you need the imaginary part of $$\int \frac{e^{iy}}{y^{s+1}}\, dy$$ Change variable $y=ix$ to get $$I=i^{-s} \int \frac {e^x}{x^{s+1}}\,dx=-i^{-s}\,\Gamma (-s,x)$$ I do not think that you could avoid it.