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You can find this formula in page 537 of Lagarias' survey: J. C. Lagarias, Euler's constant: Euler's work and modern developments, Bulletin Amer. Math. Soc. 50 (2013) 527--628.
@neverevernever Since $\lim_{x\to0} \Gamma(a,x)=\Gamma(a)$. A good bound for $x<a$ is $\Gamma(a,x)\le \Gamma(a)$. Perhaps this can be improved, but only slightly when $x$ is near $0$.
One speaks of algebras, $\sigma$-algebras of parts of a set, as algebras and $\sigma$-algebras in that set. What are your meaning? the elements of the algebra what relation have to $\omega$ in your sense?
The more general definition is by Hausdorff measure. You only need a metric space to define it. A good reference will be the book by Folland, Real Analysis, where the connection with the more elementary definitions is done.
@OneTwoOne Your reasoning is not correct. The difference between $\arg\zeta(1/2+it)$ and the other expression you equals it is not bounded. Some argument of a complex number and another one may differ in any multiple of $2\pi$. We know this is not bounded.
