Mertens function

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

The function is named in honour of Franz Mertens.

This definition can be extended to positive real numbers as follows: Less formally,

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 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, uniformly in

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(x1/2 + ε).

on the reals we have that if one assumes various conjectures about the Riemann zeta function.

A curious relation given by Mertens himself involving the second Chebyshev function is Assuming that the Riemann zeta function has no multiple non-trivial zeros, one has the "exact formula" by the residue theorem: Weyl conjectured that the Mertens function satisfied the approximate functional-differential equation where H(x) is the Heaviside step function, B are Bernoulli numbers, and all derivatives with respect to t are evaluated at t = 0.

is the Farey sequence of order n. This formula is used in the proof of the Franel–Landau theorem.

This formulation[citation needed] expanding the Mertens function suggests asymptotic bounds obtained by considering the Piltz divisor problem, which generalizes the Dirichlet divisor problem of computing asymptotic estimates for the summatory function of the divisor function.

Neither of the methods mentioned previously leads to practical algorithms to calculate the Mertens function.

Using sieve methods similar to those used in prime counting, the Mertens function has been computed for all integers up to an increasing range of x.

A combinatorial algorithm has been developed incrementally starting in 1870 by Ernst Meissel,[8] Lehmer,[9] Lagarias-Miller-Odlyzko,[10] and Deléglise-Rivat[11] that computes isolated values of M(x) in O(x2/3(log log x)1/3) time; a further improvement by Harald Helfgott and Lola Thompson in 2021 improves this to O(x3/5(log x)3/5+ε),[12] and an algorithm by Lagarias and Odlyzko based on integrals of the Riemann zeta function achieves a running time of O(x1/2+ε).

Ng notes that the Riemann hypothesis (RH) is equivalent to for some positive constant

Other upper bounds have been obtained by Maier, Montgomery, and Soundarajan assuming the RH including Known explicit upper bounds without assuming the RH are given by:[14] It is possible to simplify the above expression into a less restrictive but illustrative form as: