Consider the partial sums of the harmonic series
These partial sums actually have a special name. The partial sum is called the harmonic number, denoted by . Harmonic numbers are interesting for many reasons, one of which is their connection to the Riemann hypothesis. University of Michigan mathematician Jeffrey Lagarias proved that the Riemann hypothesis is equivalent to the inequality
for all , with the inequality being strict for . Here, denotes the sum of all divisors of .
Today, I am interested in a much more elementary question about the harmonic numbers, which I came across while perusing Peter Clark’s number theory lecture notes. Obviously, , an integer. But , , … Are there any natural numbers such that is an integer? My solution is after the jump.
The answer is no. To prove this, it suffices by the Fundamental Theorem of Arithmetic to show that, for any , can be written as
We will prove this claim by induction on . We’ve already established the base case is above, so suppose that , where , , and . Consider
If is even, then we can write , where , to obtain
Without loss of generality, suppose that , so that
But , since is odd and is even. Now suppose that is odd. Then is odd, and therefore is odd and coprime with .