Mertens’ Conjecture

January 24, 2020

Dr Holly Krieger discussed Mertens’ Conjecture on Numberphile yesterday:

Mertens’ Conjecture: The absolute value of the Mertens function M(n), computed as the sum for k from 1 to n of the Moebius function μ(k), is less than the square root of n. The Moebius function μ(n) is (-1)^k, where k is the number of prime factors of n, but 0 if n has any repeated prime factors.

The conjecture has been proved false, though no counter-examples are known. You can read more about Mertens’ Conjecture at MathWorld or Wikipedia.

Your task is to write a program to compute Mertens’ function M(n) and use it to explore some of the sequences at OEIS. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.

Pages: 1 2