The Mertens function, denoted as M(n) or m(n), is a function in number theory defined as the sum of the Möbius function μ(k) for all positive integers k up to n. Formally, it is given by: $M(n) = \sum_{k=1}^{n} \mu(k)$
where μ(k) is the Möbius function, defined as:
- μ(1) = 1
- μ(k) = 0 if k has one or more squared prime factors (i.e., k is not square-free).
- μ(k) = (-1)^r if k is the product of r distinct prime numbers (i.e., k is square-free).
The Mertens function is closely related to the distribution of prime numbers and is of significant interest in analytic number theory due to its connection with the Riemann Hypothesis. The Riemann Hypothesis is equivalent to the statement that M(n) = O(n^(1/2 + ε)) for every ε > 0, meaning that the absolute value of M(n) grows no faster than a power of n slightly larger than n^(1/2).
Historically, the Mertens Conjecture (also known as the Mertens hypothesis) stated that |M(n)| < sqrt(n) for all n > 1. This conjecture was a strong form of the Riemann Hypothesis, implying it. However, the Mertens Conjecture was disproven by Herman te Riele and Andrew Odlyzko in 1985, who showed that the inequality is violated for some n. Subsequent research refined these findings, indicating that the first violation occurs for a very large n (around exp(3.21 x 10^6)). While the conjecture itself is false, the study of the growth rate of M(n) remains crucial.
The Mertens function is known to fluctuate around zero, and its values can be both positive and negative. Although the Mertens Conjecture was disproven, it is still believed that M(n) grows slower than any positive power of n, which is a weaker statement but still consistent with the Riemann Hypothesis. The precise behavior of M(n) provides insights into the irregularities of the distribution of square-free numbers and the zeros of the Riemann zeta function.