Horizontal Monotonicity of the Modulus of the Riemann Zeta Function and   Related Functions
  It is shown that the absolute values of Riemann's zeta function and two related functions strictly decrease when the imaginary part of the argument is fixed to any number with absolute value at least 8 and the real part of the argument is negative and increases up to 0; extending this monotonicity to the increase of the real part up to 1/2 is shown to be equivalent to the Riemann Hypothesis. Another result is a double inequality relating the real parts of the logarithmic derivatives of the three functions under consideration. 
Yuri Matiyasevich, Filip Saidak, Peter Zvengrowski
math.NT math.CV