Mertens Conjecture: A Fabulous Mathematical Contradiction

Some mathematical problems can be solved in a reasonable amount of time. Others demand entire decades, and despite the final result, you never manage to shake the feeling that something is wrong with the whole process. One of the most striking examples is the so-called Mertens conjecture, which dates back to the late 19th century. Experts needed a hundred years to prove that the conjecture is false, but all the available evidence we have (and will have in the future) suggests the opposite...

So how does that work exactly? In general terms, it "seems true but is false"; however, any serious mathematician would throw a shoe at our head if we stopped there. The story begins with a doubt: If you take a random number, is it more likely to have an even number of prime factors, or an odd number?

For example, take the case of 10: if we split it into prime factors, we get 2 × 5, so the number of factors is even. In contrast, if we focus on any prime number, the number of prime factors is automatically odd. But there is an additional condition we cannot ignore: when prime factors repeat (e.g., 12 = 2 × 2 × 3). That brings us to the first half of the Mertens conjecture, with the function μ(n). As Dr. Holly Krieger explains in the video, there are three possibilities for μ(n):

1. If n has an even number of non-repeated prime factors, the value is 1.
2. If n has an odd number of non-repeated prime factors, the value is -1.
3. If n has repeated factors, the value is 0.

The rest is a matter of projection: the value of n for 1 is 1, for 2 is -1, for 3 is -1 again, for 4 is 0 (because 2 × 2 repeats factors), for 5 is -1, for 6 is 1, and so on... but we still must answer the original question: even factors, or odd factors? To find the answer, the idea is to sum all the values of n and see which side the total M(n) falls on, whether positive (even) or negative (odd).

The Second Half of the Conjecture

And here is when the second half of the conjecture makes its appearance: in the late 19th century, the mathematician Thomas Joannes Stieltjes sent a letter to his colleagues Charles Hermite and Franz Mertens stating that M(n) (today known as the Mertens function) can never be larger than the square root of n, whether negative or positive. If we pour all the computational firepower we have into a graph, we will see that M(n) oscillates almost chaotically, but it seems to respond correctly to Stieltjes's announcement:

The problem is... that the conjecture is false. At a very, very, very far end of the graph, M(n) crosses the square root barrier, no matter how much all our calculations suggest the opposite. Mathematicians Andrew Odlyzko and Herman te Riele proved in 1985, with the help of the Lenstra–Lenstra–Lovász lattice basis reduction algorithm (or "LLL"), that the conjecture breaks down at the level of 10^(10^23)... a number so large that we don't have enough atoms in the Universe to represent it. At the same time, if the Mertens conjecture were true, that would imply that the famous Riemann hypothesis is also true, a very interesting story to explore... another time.

Source: Numberphile on YouTube