journal-of-discrete-mathematical-sciences-and-cryptography Journal of Discrete Mathematical Sciences and Cryptography 2169-0065 0972-0529 Taru Publications 10.47974/JDMSC-2414 On some specific order estimates for the Mertens function and its relation to the non-trivial zeros of ζ(s) De Subham Department of Mathematics, Hauz Khas, Indian Institute of Technology Delhi, New Delhi, Delhi, 110016, India 29 7 2757 2772 12 03 2026

This article presents a brief overview of the Mertens function M(n) and Redheffer matrices An, deriving specific order estimates for M(n) to investigate the distribution of non-trivial zeros of the Riemann zeta function ζ(s). Using these tools, we establish a necessary and sufficient condition for the validity of the Riemann Hypothesis.

 Arithmetic function Mö̈bius function Riemann zeta function Mertens function Redheffer matrix Riemann hypothesis Roesler matrix Spectral radius Mertens hypothesis access open retracted no