Mertens

Mertens is a surname of Dutch and Flemish origin. It is most common in Belgium and the Netherlands and among Dutch-speaking communities abroad. The name is typically a patronymic from a given name such as Mert or Merte, and it has several variants, including Merten and Mertens.

Notable people with the surname include Dries Mertens (born 1987), a Belgian professional footballer who has played as a forward or attacking midfielder for clubs in Italy and for the Belgian national team. The surname is associated with various individuals across different fields, reflecting its broad distribution in Dutch-speaking regions.

In mathematics, the Mertens function M(x) is defined as the summatory Möbius function, M(x) = sum_{n ≤ x} μ(n), where μ(n) is the Möbius function. It is a central object in analytic number theory and is connected to the Mertens theorems, which describe the average behavior of arithmetic functions and relate to questions about prime number distribution. There is also the Mertens conjecture, which posits that |M(x)| ≤ sqrt(x) for all x; this conjecture was disproven in 1985 by Odlyzko and te Riele. The Mertens function is named after the Belgian mathematician Franz Mertens, who contributed to the study of these topics.