Hi everyone!

In this blog, we will learn how to find the asymptotic expansion of $$$H_n = \sum\limits_{k=1}^n \frac{1}{k}$$$.

Motivational example

In the recent Meta Hacker Cup, problem C of the round 3 reduced to finding the sum of $$$\frac{1}{a+bk}$$$ for given $$$a$$$ and $$$b$$$, over all $$$k \in [L, R)$$$.

Now, putting it into Wolfram, you can see that

where $$$\psi(z)$$$ is the digamma function. It is sufficient to solve this problem, as you can use boost or scipy to find it. But I feel like it's long time to satiate my curiosity on how it can be computed without knowing the digamma function in advance, and this blog is focused just on that.

Harmonic numbers

First of all, let's define $$$H(n) = \sum\limits_{k=1}^{n} \frac{1}{k}$$$, the $$$n$$$-th harmonic number. If $$$a = 0$$$, we can say that

Notice how it is similar to the formula with the digamma function? That's not a coincidence! In fact, we can e.g. re-define $$$H(n)$$$ as

and this definition will be quite easy to update to also allow non-integer $$$n$$$, by changing the upper sumation limit to $$$\lfloor n \rfloor - 1$$$.

With this in mind, we can re-express the whole sum as

This is also similar to the formula with $$$\psi$$$, because $$$\psi(n) = H(n-1)-\gamma$$$ for integer $$$n$$$, where $$$\gamma$$$ is the Euler–Mascheroni constant.

Asymptotics of $$$H(n)$$$

As a very rough first estimate, we can approximate

which is a quite known estimation in many applications, e.g. when finding all divisors of all numbers up to $$$n$$$.

Now, the approach that we will use to finding the true asymptotic expansion of $$$H(n)$$$ is assuming that $$$H(n) = \ln n + F(\frac{1}{n})$$$, where $$$F(x)$$$ is an analytic function, meaning that it can be expressed as $$$F(x) = a_0 + a_1 x + a_2 x^2 + \dots$$$, i.e. as a power series.

Finding the expansion of $$$F(\frac{1}{n})$$$

With this assumption, let's find the coefficients of $$$F(x)$$$. For this, consider

In this form, we can rewrite the LHS as a power series in $$$\frac{1}{n}$$$. Indeed,

as follows from the standard expansion $$$\log \frac{1}{1-x} = \sum\limits_{k=1}^\infty \frac{x^k}{k}$$$. At the same time,

can also be represent as a power series in $$$\frac{1}{n}$$$ as

as follows from the standard expansion for $$$\frac{1}{(1-x)^k}$$$. Putting it all back together, we arrive at

For $$$k=1$$$, the coefficients near $$$\frac{1}{n^k}$$$ on both sides are already same, and for all higher $$$k$$$ this requirement turns into

Using the identity above for $$$k=2,3,\dots$$$ allows us to find $$$a_1 = \frac{1}{2}$$$, $$$a_2 = -\frac{1}{12}$$$, $$$a_3=0$$$, $$$a_4=\frac{1}{120}$$$, and so on. Okay but what about $$$a_0$$$? Well, actually, $$$a_0=\gamma$$$! But it is not important for our purposes at all, because it cancels out when we subtract $$$H(L-1)$$$ from $$$H(R-1)$$$.

Relation to Riemann zeta function

One can also derive the same result in the form

where $$$\zeta(x)$$$ is the Riemann zeta-function that already occurred e.g. here, but I fail to give any meaningful interpretation to this fact.

Afterthoughts

Are there any simpler ways to derive it?