Editing Mertens conjecture (section)

{{Short description|Disproved mathematical conjecture}}
[[Image:Congettura Mertens.png|thumb|right|The graph shows the [[Mertens function]] <math>M(n)</math> and the square roots <math>\pm \sqrt{n}</math> for <math>n \le 10,000</math>. After computing these values, Mertens conjectured that the absolute value of <math>M(n)</math> is always bounded by <math>\sqrt{n}</math>. This hypothesis, known as the Mertens conjecture, was disproved in 1985 by [[Andrew Odlyzko]] and [[Herman te Riele]].]]

In [[mathematics]], the '''Mertens conjecture''' is the statement that the [[Mertens function]] <math>M(n)</math> is bounded by <math>\pm\sqrt{n}</math>. Although now disproven, it had been shown to imply the [[Riemann hypothesis]].  It was conjectured by [[Thomas Joannes Stieltjes]], in an 1885 letter to [[Charles Hermite]] (reprinted in {{harvs|txt|last=Stieltjes|authorlink=Thomas Joannes Stieltjes|year=1905}}), and again in print by {{harvs|txt|authorlink=Franz Mertens|last=Mertens|first=Franz|year=1897}}, and disproved by {{harvs|txt|last1=Odlyzko|first1=Andrew|authorlink1=Andrew Odlyzko|last2=te Riele|first2=Herman|authorlink2=Herman te Riele|year=1985}}.
It is a striking example of a mathematical conjecture proven false despite a large amount of computational evidence in its favor.