, for all x

To phrase it in another way, the absolute value of M(x) would always be less than the square root of x. Mertens would then calculate values for M(x) up to x = 10,000 and made his conjecture after staring hard at his list of calculations.

In that same year, mathematician R. D. von Sterneck took it a step further and conjectured that  after he calculate values for M(x), this time for x running up to five million. While it was taxing on him, he delighted in finding out that  was always true after the first two hundred values.

However, his joy was short lived when years later, Sterneck conjecture was discovered to fail. For , the first time  is at , discovered in 1960 by Wolfgang Jurkat. It was way deep in the sequence but that one value was enough to disproved Sterneck’s conjecture. Focus was turned back to the original Mertens conjecture and it was still held valid for x up to 7.8 billion.

Sadly, the Mertens conjecture would not stand the test of time when in 1983, Herman te Riele and Andrew Odlyzko disproved that  for all x. They found the actual value of x where  is when , number to big for me to write.

This would be only one counterexample to the Mertens conjecture. The location of the first counterexample is still a mystery. Mathematicians have narrowed down the search. In 1987, J. Pintz shows that another counterexample could be found for . Riele and Odlyzko believed that no counterexamples existed for . This gives mathematicians the range of  to find the first counterexample to disprove Mertens conjecture.

It’s amazing to see how a small little function can puzzle even the greatest of mathematicians. It also goes to show that functions are in fact not boring, and if one were to read between the numbers, there is a wealth of clues waiting to be discovered answering the purpose of the function.