On an inequality for the Riemann zeta-function in the critical strip

2012/06/08 by Nazardonyavi, Sadegh, Yakubovich, Semyon
#11M26 #11M99 #26Dxx #33B15 #41A17 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1206.1801

Abstract

By using new power inequalities we give an elementary proof of an important relation for the Riemann zeta-function |ζ(1-s)| <= |ζ(s)| in the strip 0< Re s<1/2, |\Im s| >= 12. Moreover, we establish a sufficient condition of the validity of the Riemann hypothesis in terms of the derivative with respect to Re s of |ζ(s)|² and conjecture its necessity.

Related

• An investigation of the non-trivial zeros of the Riemann zeta function
• Bounding the log-derivative of the zeta-function
• On zeros of the Riemann zeta function
• An analytic approach to the Riemann hypothesis
• A remark on density theorems for Riemann's zeta-function