Mertens function

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In number theory, the Mertens function is defined for all positive integers n as

${\displaystyle M(n)=\sum _{k=1}^{n}\mu (k),}$

where ${\displaystyle \mu (k)}$ is the Möbius function. The function is named in honour of Franz Mertens. This definition can be extended to positive real numbers as follows:

${\displaystyle M(x)=M(\lfloor x\rfloor ).}$

Less formally, ${\displaystyle M(x)}$ is the count of square-free integers up to x that have an even number of prime factors, minus the count of those that have an odd number.

The first 143 M(n) values are (sequence A002321 in the OEIS)

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 values

2, 39, 40, 58, 65, 93, 101, 145, 149, 150, 159, 160, 163, 164, 166, 214, 231, 232, 235, 236, 238, 254, 329, 331, 332, 333, 353, 355, 356, 358, 362, 363, 364, 366, 393, 401, 403, 404, 405, 407, 408, 413, 414, 419, 420, 422, 423, 424, 425, 427, 428, ... (sequence A028442 in the OEIS).

Because the Möbius function only takes the values −1, 0, and +1, the Mertens function moves slowly, and there is no x such that |M(x)| > x. H. Davenport^[1] demonstrated that, for any fixed h,

${\displaystyle \sum _{n=1}^{x}\mu (n)\exp(i2\pi n\theta )=O\left({\frac {x}{\log ^{h}x}}\right)}$

uniformly in ${\displaystyle \theta }$. This implies, for ${\displaystyle \theta =0}$ that

${\displaystyle M(x)=O\left({\frac {x}{\log ^{h}x}}\right)\ .}$

The Mertens conjecture went further, stating that there would be no x where the absolute value of the Mertens function exceeds the square root of x. 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(x), namely M(x) = O(x^{1/2 + ε}). Since high values for M(x) grow at least as fast as ${\displaystyle {\sqrt {x}}}$, this puts a rather tight bound on its rate of growth. Here, O refers to big O notation.

The true rate of growth of M(x) is not known. An unpublished conjecture of Steve Gonek states that

${\displaystyle 0<\limsup _{x\to \infty }{\frac {|M(x)|}{{\sqrt {x}}(\log \log \log x)^{5/4}}}<\infty .}$

Probabilistic evidence towards this conjecture is given by Nathan Ng.^[2] In particular, Ng gives a conditional proof that the function ${\displaystyle e^{-y/2}M(e^{y})}$ has a limiting distribution ${\displaystyle \nu }$ on ${\displaystyle \mathbb {R} }$. That is, for all bounded Lipschitz continuous functions ${\displaystyle f}$ on the reals we have that

${\displaystyle \lim _{Y\to \infty }{\frac {1}{Y}}\int _{0}^{Y}f{\big (}e^{-y/2}M(e^{y}){\big )}\,dy=\int _{-\infty }^{\infty }f(x)\,d\nu (x),}$

if one assumes various conjectures about the Riemann zeta function.