Horizontal monotonicity of the modulus of the zeta function, \(L\)-functions, and related functions (Q2928543)

The authors prove the assertion stating that \(\left| \zeta \left( \sigma +it\right) \right| \) is strictly decreasing in \(\sigma \) for \(0<\sigma < \frac{1}{2}\) is equivalent to the Riemann hypothesis. They also show that \( \left| \zeta \left( s\right) \right| \) is decreasing in \(\sigma \) for \(\sigma <0\). These two results are valid for \(\left| t\right| \geqslant 8\). The authors further consider the corresponding results for \(\eta \left( s\right) =\left( 1-2^{1-s}\right) \zeta \left( s\right) \) and Dirichlet \(L\)-function \(L\left( s,\chi \right) \) and the related \( L\)-function \(\xi \left( s,\chi \right) \). They conclude the paper by exploring the relation to the Selberg class of degree 1.