Horizontal Monotonicity of the Modulus of the Riemann Zeta Function and Related Functions

TL;DR

This paper investigates the monotonic behavior of the modulus of the Riemann zeta function and related functions in the critical strip, linking their properties to the Riemann Hypothesis and providing inequalities for their derivatives.

Contribution

It establishes strict monotonicity of these functions' absolute values in certain regions and connects this to the Riemann Hypothesis, offering new inequalities for their derivatives.

Findings

Absolute values decrease with increasing real part in specified regions.

Monotonicity extension is equivalent to the Riemann Hypothesis.

Derived inequalities relate the logarithmic derivatives of the functions.

Abstract

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.