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Please help me on the following.

I need to estimate $$\int_{1-\frac{c}{\log t}-iT}^{1-\frac{c}{\log t}+iT}\frac{\zeta(s-1)}{\zeta(s)}\frac{x^s}{s}ds$$ where $c$ is a constant, $T>0$.

What i tried to use is Functional equation $\zeta(s)\Gamma(\frac s2)=\pi^{\frac12-s} \frac{\Gamma(\frac{1-s}{2})}{\Gamma(\frac{s}{2})}\zeta(1-s)$. BUt I have no idea how shall i proceed.

KON3
  • 4,111
  • When I see an expression type $1-c/\log t$ i think about zero-free region. Make no sense that this $t$ is the same that appears in $s=\sigma+i t$, because we are integrating $s$, so what does we know about this $t$? There are zeros of $\zeta$ on the right of the line of integration? – MrSelberg Aug 30 '15 at 10:02
  • I was asking because I did an estimate, but assumes that there are no zeros to the right... – MrSelberg Sep 03 '15 at 03:08

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