Odlyzko, A. M.; te Riele, Herman J. J. Disproof of the Mertens conjecture. (English) Zbl 0544.10047 J. Reine Angew. Math. 357, 138-160 (1985). The Mertens conjecture states that \(| M(x)|<x^{1/2}\) for all \(x>1\), where \(M(x)=\sum_{n\leq x}\mu(n)\), and \(\mu\) (n) is the Möbius function. This conjecture has attracted a substantial amount of interest in its almost 100 years of existence because its truth was known to imply the truth of the Riemann hypothesis. This paper disproves the Mertens conjecture by showing that \[ \lim \sup_{x\to \infty}M(x)x^{- 1/2}\quad>\quad 1.06,\quad and\quad \lim \inf_{x\to \infty}M(x)x^{- 1/2}\quad<\quad -1.009. \] The disproof relies on extensive computations with the zeros of the zeta-function, and does not provide an explicit counterexample. Cited in 18 ReviewsCited in 61 Documents MSC: 11N37 Asymptotic results on arithmetic functions 11M06 \(\zeta (s)\) and \(L(s, \chi)\) 11-04 Software, source code, etc. for problems pertaining to number theory Keywords:disproof; computation of zeros of Riemann zeta-function; Mertens conjecture; Möbius function PDFBibTeX XMLCite \textit{A. M. Odlyzko} and \textit{H. J. J. te Riele}, J. Reine Angew. Math. 357, 138--160 (1985; Zbl 0544.10047) Full Text: DOI EuDML Online Encyclopedia of Integer Sequences: Mertens’s function: Sum_{k=1..n} mu(k), where mu is the Moebius function A008683. Nearest integer to imaginary part of n-th zero of Riemann zeta function. From Mertens’s conjecture (1): floor(sqrt(n)) - |M(n)|, where M is Mertens’s function A002321. From Mertens’s conjecture (2): floor(sqrt(n)) - Mertens’s function A002321(n).