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Encyclopedia > Mertens function

In number theory, the Mertens function is

where μ(k) is the M�bius function.

Because the M�bius 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 . 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. Results from FactBites:

 Mertens function - Wikipedia, the free encyclopedia (294 words) 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. Mertens, "Über eine zahlentheoretische Funktion", Akademie Wissenschaftlicher Wien Mathematik-Naturlich Kleine Sitzungsber, IIa 106, (1897) 761-830.
 Möbius function (527 words) The classic Möbius function μ(n) is an important multiplicative function considered in number theory and in combinatorics. The Möbius function is multiplicative and is of relevance in the theory of multiplicative and arithmetic functions because it appears in the Möbius inversion formula. This function is closely linked with the positions of zeroes of the Euler - Riemann ζ- function.
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