I Does the Integral of Riemman Zeta Function have a meaning?

[JorgeM]

I have been trying to use numerical methods with this function but now I realize that I if I could suggest a Polynomial in theory, I could get some value for the Integral at least in any interval. In general, does the Integral of the Riemman dseta function has a meaning by itself?

 

[martinbn]

Which integral?

 

[aheight]

Perhaps this one:

$$ \zeta(s)=\frac{\Gamma(1-s)}{2 \pi i} \mathop\int_{\multimap} \frac{x^{s-1}}{e^{-x}-1}dx$$

Nevertheless, one of the most beautiful constructs in Mathematics IMO.

 

[lavinia]

I have been trying to use numerical methods with this function but now I realize that I if I could suggest a Polynomial in theory, I could get some value for the Integral at least in any interval. In general, does the Integral of the Riemman dseta function has a meaning by itself?

I think that the Riemann zeta function is meromorphic in the entire complex plane with a single pole at 1 . It makes sense to takes its line integral along any piecewise smooth curve that goes not pass through the pole.