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

\( M(n) = \sum_{k=1}^n \mu(k) \)

where 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 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, ... (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:

\( M(x) = \sum_{1\le k \le x} \mu(k). \)

Representations
As an integral

Using the Euler product one finds that

\( \frac{1}{\zeta(s) }= \prod_{p} (1-p^{-s})= \sum_{n=1}^{\infty} \frac{\mu(n)}{n^s} \)

where \( \zeta(s) \) is the Riemann zeta function and the product is taken over primes. Then, using this Dirichlet series with Perron's formula, one obtains:

\( \frac{1}{2\pi i}\oint_{C} \frac{x^{s}}{s\zeta(s)} \, ds = M(x) \)

where C is a closed curve[disambiguation needed ] encircling all of the roots of \zeta(s).

Conversely, one has the Mellin transform

\( \frac{1}{\zeta(s)} = s\int_1^\infty \frac{M(x)}{x^{s+1}}\,dx \)

which holds for \( \mathrm{Re}(s)>1. \)

A curious relation given by Mertens himself involving the second Chebyshev function is

\( \Psi (x) = M\left( \frac{x}{2} \right) \log(2)+M \left( \frac{x}{3} \right) \log(3) + M \left( \frac{x}{4}\right )\log(4) + \cdots. \)

A good evaluation, at least asymptotically, would be to obtain, by the method of steepest descent, the inequality

\( \oint_C F(s)e^{st} \, ds \sim M(e^{t}). \)

Assuming that there are not multiple non-trivial roots of \zeta (\rho) we have the "exact formula" by the residue theorem:

\( \frac{1}{2 \pi i} \oint_C \frac{x^s}{s \zeta (s)} \, ds = \sum_\rho \frac{x^\rho}{\rho \zeta'(\rho)} - 2+\sum_{n=1}^\infty \frac{ (-1)^{n-1} (2\pi )^{2n}}{(2n)! n \zeta(2n+1)x^{2n}}. \)

Weyl conjectured that the Mertens function satisfied the approximate functional-differential equation

\( \frac{y(x)}{2}-\sum_{r=1}^N \frac{B_{2r}}{(2r)!}D_t^{2r-1} y \left(\frac{x}{t+1}\right) + x\int_0^x \frac{y(u)}{u^{2}} \, du = x^{-1}H(\log x) \)

where H(x) is the Heaviside step function, B are Bernoulli numbers and all derivatives with respect to t are evaluated at t = 0.

Titchmarsh (1960) provided a Trace formula involving a sum over the Möbius function and zeros of Riemann Zeta in the form

\( \sum_{n=1}^{\infty}\frac{\mu(n)}{\sqrt{n}} g \log n = \sum_t \frac{h(t)}{\zeta'(1/2+it)}+2\sum_{n=1}^\infty \frac{ (-1)^{n} (2\pi )^{2n}}{(2n)! \zeta(2n+1)}\int_{-\infty}^{\infty}g(x) e^{-x(2n+1/2)} \, dx, \)

where 't' sums over the imaginary parts of nontrivial zeros, and (g, h) are related by a Fourier transform, such that

\( \pi g(x)= \int_{0}^{\infty}h(u)\cos(ux) \, du. \)

As a sum over Farey sequences

Another formula for the Mertens function is

\( M(n)= \sum_{a\in \mathcal{F}_n} e^{2\pi i a} \) where \( \mathcal{F}_n \) is the Farey sequence of order n.

This formula is used in the proof of the Franel–Landau theorem.[1]
As a determinant

M(n) is the determinant of the n × n Redheffer matrix, a (0,1) matrix in which aij is 1 if either j is 1 or i divides j.
Calculation

The Mertens function has been computed for an increasing range of n.

Person	Year	Limit
Mertens	1897	10⁴
von Sterneck	1897	1.5×10⁵
von Sterneck	1901	5×10⁵
von Sterneck	1912	5×10⁶
Neubauer	1963	10⁸
Cohen and Dress	1979	7.8×10⁹
Dress	1993	10¹²
Lioen and van de Lune	1994	10¹³
Kotnik and van de Lune	2003	10¹⁴

The Mertens function for all integer values up to N may be computed in O(N1+ε) time. Elementary algorithms exist to compute isolated values of M(N) in O(N2/3+ε) time.

See OEIS A084237 for values of M(N) at powers of 10.
Notes

^ Edwards, Ch. 12.2