Graphs of numerical examples show that in the critical strip: modulus(Zeta(x+it) < modulus[Zeta(it)]. The question concerns the possibility of generalizing this inequality. In "Lectures on the Riemann Zeta Function: http://www.math.tifr.res.in/~publ/ln/tifr01.pdf , it was proved that: Modulus[Zeta(x + it)] < t^(1-e) *[ 1+ 3/e + 1/(1-e)] for x >= e,where 0 < e < 1 since 1/(1-e) > 0, Modulus[Zeta(x + it)] < t^(1-e) *[ 1 + 1/(1-e)]
Therefore: for x = 0, e = 0, Modulus[Zeta(it)] < 2*t Hence: Modulus[Zeta(x + it)]/Modulus[Zeta(it)] < 1 since Modulus[Zeta(it)] > 0
Whence: Modulus[Zeta(x + it)] < Modulus[Zeta(it)]
If this reasoning is not correct, is there another way to prove the inequality above?
Thank you