Euler-Mascheroni Constant from Zeta: Im{z(1+it)} v. Re{z(1+it)}

Using Eq. 21 from mathworld.wolfram.com/RiemannZetaFunction.html (shown in the inset at upper left) it can be shown graphically that the Euler-Mascheroni constant (gamma) is the real part of the Riemann zeta function at s = 1 + it as t approaches 0.

 

This can easily be proved by letting s = 1 + it in eq. 22 from mathworld.wolfram.com/RiemannZetaFunction.html , then taking the real part of the result and letting x ~ sin(x) for small x. The result is eq. 56 from mathworld.wolfram.com/Euler-MascheroniConstant.html

 

Done