Does the Integral of Riemman Zeta Function have a meaning?

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SUMMARY

The integral of the Riemann zeta function, represented as $$ \zeta(s)=\frac{\Gamma(1-s)}{2 \pi i} \mathop\int_{\multimap} \frac{x^{s-1}}{e^{-x}-1}dx$$, holds significance in mathematical analysis, particularly when evaluated along piecewise smooth curves that do not intersect its pole at 1. Numerical methods can be employed to approximate values of this integral over specified intervals. The Riemann zeta function is meromorphic across the complex plane, reinforcing its relevance in complex analysis.

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JorgeM
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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?
 
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Which integral?
 
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.
 
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JorgeM said:
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.
 

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