Mertens function

Article summary

In number theory, the Mertens function is defined for all positive integers n as : M(n) = \sum_{k=1}^n \mu(k), where \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: : M(x) = M(\lfloor x \rfloor). Less formally, 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 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, ... . 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 demonstrated that, for any fixed h, : \sum_{n=1}^{x}\mu(n)\exp(i2\pi n\theta) = O\left(\frac{x}{\log^hx}\right) uniformly in \theta . This implies, for \theta=0 that : M(x) = O\left(\frac{x}{\log^hx}\right)\ .…

Key claims

No claims recorded.

Discussion questions

1. Summarise the article in your own words. Which single claim would you verify first, and why?
2. Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
3. If two sources disagreed about one of these facts, what would you need in order to trust one over the other?

Bibliography

No sources recorded.