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- Mertens_function abstract "In number theory, the Mertens function is defined for all positive integers n aswhere μ(k) is the Möbius function. The function is named in honour of Franz Mertens.Less formally, M(n) is the count of square-free integers up to n that have an even number of prime factors, minus the count of those that have an odd number.The first 160 M(n) is:The Mertens function slowly grows in positive and negative directions both on average and in peak value, oscillating in an apparently chaotic manner passing through zero when n has the values2, 39, 40, 58, 65, 93, 101, 145, 149, 150, 159, 160, 163, 164, 166, 214, 231, 232, 235, 236, 238, 254, ... (sequence A028442 in OEIS).Because the Möbius function only takes the values −1, 0, and +1, the Mertens function moves slowly and there is no n such that |M(n)| > n. The Mertens conjecture went further, stating that there would be no n where the absolute value of the Mertens function exceeds the square root of n. The Mertens conjecture was proven false in 1985 by Andrew Odlyzko and Herman te Riele. However, the Riemann hypothesis is equivalent to a weaker conjecture on the growth of M(n), namely M(n) = O(n1/2 + ε). Since high values for M(n) grow at least as fast as the square root of n, this puts a rather tight bound on its rate of growth. Here, O refers to Big O notation.The above definition can be extended to real numbers as follows:".
- Mertens_function thumbnail Mertens.svg?width=300.
- Mertens_function wikiPageExternalLink 1047565447.
- Mertens_function wikiPageExternalLink mertens.disproof.pdf.
- Mertens_function wikiPageExternalLink 42.
- Mertens_function wikiPageID "435639".
- Mertens_function wikiPageRevisionID "605241107".
- Mertens_function date "December 2009".
- Mertens_function hasPhotoCollection Mertens_function.
- Mertens_function name "Mertens's function".
- Mertens_function sequencenumber "A002321".
- Mertens_function talk "y".
- Mertens_function title "Mertens function".
- Mertens_function urlname "MertensFunction".
- Mertens_function subject Category:Arithmetic_functions.
- Mertens_function type Abstraction100002137.
- Mertens_function type ArithmeticFunctions.
- Mertens_function type Function113783816.
- Mertens_function type MathematicalRelation113783581.
- Mertens_function type Relation100031921.
- Mertens_function comment "In number theory, the Mertens function is defined for all positive integers n aswhere μ(k) is the Möbius function.".
- Mertens_function label "Fonction de Mertens".
- Mertens_function label "Función de Mertens".
- Mertens_function label "Funkcja Mertensa".
- Mertens_function label "Funzione di Mertens".
- Mertens_function label "Função de Mertens".
- Mertens_function label "Mertens function".
- Mertens_function label "Mertensfunctie".
- Mertens_function label "Функция Мертенса".
- Mertens_function label "دالة ميرتنز".
- Mertens_function label "梅滕斯函數".
- Mertens_function sameAs Mertensova_funkce.
- Mertens_function sameAs Función_de_Mertens.
- Mertens_function sameAs Fonction_de_Mertens.
- Mertens_function sameAs Funzione_di_Mertens.
- Mertens_function sameAs 메르텐스_함수.
- Mertens_function sameAs Mertensfunctie.
- Mertens_function sameAs Funkcja_Mertensa.
- Mertens_function sameAs Função_de_Mertens.
- Mertens_function sameAs m.0287b6.
- Mertens_function sameAs Q841254.
- Mertens_function sameAs Q841254.
- Mertens_function sameAs Mertens_function.
- Mertens_function wasDerivedFrom Mertens_function?oldid=605241107.
- Mertens_function depiction Mertens.svg.
- Mertens_function isPrimaryTopicOf Mertens_function.
