Mertens function

From Academic Kids

In number theory, the Mertens function is

<math>M(n) = \sum_{1\le k \le n} \mu(k)<math>

where μ(k) is the Möbius function.

Because the Möbius function has only the return values -1, 0 and +1, it's obvious that the Mertens function moves slowly and that there is no x such that M(x) > x. The Mertens conjecture goes even further, stating that there is no x where the absolute value of the Mertens function exceeds the square root of x. The Mertens conjecture was disproven in 1985. However, the Riemann hypothesis is equivalent to a weaker conjecture on the growth of M(x), namely <math>M(x) = o(x^{\frac12 + \epsilon})<math>. Since high values for M grow at least as fast as the square root of x, this puts a rather tight bound on its rate of growth.

External links

  • Values of the Mertens function for the first 50 n are given by SIDN A002321 (http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A002321)
  • Values of the Mertens function for the first 2500 n are given by PrimeFan's Mertens Values Page (http://www.geocities.com/primefan/Mertens2500.html)es:Funcin de Mertens

fr:Fonction de Mertens ko:메르텐스 함수 it:Funzione di Mertens nl:Mertensfunctie sl:Mertensova funkcija

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