Eckhard Müller-Mertens (1923–2015)

The Mertens Conjecture Revisited

Disproof of the Mertens Conjecture

The Mertens conjecture states that  M(x)  < x ⁄2 for all x > 1, where M(x) = n ≤ x Σ μ(n) , and μ(n) is the Mo bius 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 x → ∞ lim sup M(x) x − ⁄2 > 1. 06 ....

M ay 2 00 5 Mertens ’ Proof of Mertens ’ Theorem Mark B . Villarino

We study Mertens’ own proof (1874) of his theorem on the sum of the reciprocals of the primes and compare it with the modern treatments.

Symmetric matrices related to the Mertens function

In this paper we explore a family of congruences over N∗ from which a sequence of symmetric matrices related to the Mertens function is built. From the results of numerical experiments we formulate a conjecture, about the growth of the quadratic norm of these matrices, which implies the Riemann hypothesis. This suggests that matrix analysis methods may play a more important role in this classic...

Improved results on the Mertens conjecture

In this article, we study the Mertens conjecture. We revisit and improve the original constructive disproof of János Pintz. We obtain a new lower bound for the minimal counterexample and new numerical results for this conjecture.