Mertens conjecture

In mathematics, the Mertens conjecture was a conjecture that the Mertens function M(n) would be bounded by √n. It was raised by Franz Mertens in 1897 and was shown to be false in 1985. The conjecture, if proven true, would have implied the truth of the Riemann hypothesis.

Definition

In number theory, the Mertens function is defined as:

M(n)=␡ ␡ 1≤ ≤ k≤ ≤ nμ μ (k){displaystyle M(n)=sum _{1leq kleq n}mu (k)}}

where μ(k) is the Möbius function, then the Mertens conjecture states that:

<math alttext="{displaystyle left|M(n)right|日本語M(n)日本語.n{displaystyle leftINDM(n)right implies{sqrt {n}}}<img alt="{displaystyle left|M(n)right|

History

In 1885, Stieltjes claimed to have proved this result, but did not publish a proof, probably because he discovered that it was in error. The conjecture was initially postulated by Franz Mertens in 1897, based on Stieltjes's partial results, he published a paper in which he opined that "it was probably true ".

However, in 1985, te Riele and Odlyzko proved that the Mertens conjecture is false.

The Mertens conjecture is interesting, because if it had been proven true, that would have implied that the famous Riemann hypothesis was also true.

Connection with the Riemann Hypothesis

The link with the Riemann hypothesis is based on the fact that the result can be derived

1γ γ (z)=z∫ ∫ 1∞ ∞ M(x)xz+1dx{displaystyle {frac {1}{zeta (z)}}=zint _{1^}{infty }{frac {M(x)}{x^{z+1}}}}}dx}

where ζ(z) is the Riemann zeta function. The Mertens conjecture would mean that this integral converges for Re(z) > 1/2, which in turn would imply that 1/ζ(z) is defined for Re(z) > 1/2 and by symmetry for Re(z) < 1/2. Thus, the only zeros of ζ(z) would be at Re(z) = 1/2, as the Riemann hypothesis says.