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Mertens function to n=10000

Mertens function to n=10,000,000

In number theory, the Mertens function is

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

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. M(n) = 0 for the n 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 has only the values −1, 0, and +1, it is obvious that the Mertens function moves slowly and that there is no k such that |M(k)| > k. The Mertens conjecture went even further, stating that there would be no k where the absolute value of the Mertens function exceeds the square root of k. The Mertens conjecture was disproven in 1985. However, the Riemann hypothesis is equivalent to a weaker conjecture on the growth of M(k), namely $M(k) = O(k^{\frac12 + \varepsilon})$ . Since high values for M grow at least as fast as the square root of k, this puts a rather tight bound on its rate of growth. Here, O refers to Big O notation.

1 Representations
2 Calculation
3 Notes
4 References

Representations

As an integral

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Using the Euler product one finds that

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

where $ζ(s)$ 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 encircling all of the roots of $ζ(s).$

Conversely, one has the Mellin transform

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

which holds for $Re(s) > 1$ .

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, an inequality:

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

assuming that there are not multiple non-trivial roots of $ζ(ρ)$ you have the "exact formula" by 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 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 mobius 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 so

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

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 a_ij 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 der Lune	1994	10¹³
Kotnik and van der Lune	2003	10¹⁴

Notes

^ Edwards, Ch. 12.2

References

Edwards, Harold (1974). Riemann's Zeta Function. Mineola, New York: Dover. ISBN 0-486-41740-9.
F. Mertens, "Über eine zahlentheoretische Funktion", Akademie Wissenschaftlicher Wien Mathematik-Naturlich Kleine Sitzungsber, IIa 106, (1897) 761–830.
A. M. Odlyzko and H.J.J. te Riele, "Disproof of the Mertens Conjecture", Journal für die reine und angewandte Mathematik 357, (1985) pp. 138–160.
Weisstein, Eric W., "Mertens function" from MathWorld.
Values of the Mertens function for the first 10,000 n are given by A002321.

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