About 2.5 years ago I had promised Joseph Nebus that I will write about the interplay between Bernoulli numbers and Riemann zeta function. In this post I will discuss a problem about finite harmonic sums which will illustrate the interplay.

Consider the Problem 1.37 from The Math Problems Notebook:

Let be a set of natural numbers such that , and are not prime numbers. Show that

Since each is a composite number, we have for some, not necessarily distinct, primes and . Next, implies that . Therefore we have:

Though it’s easy to show that , we desire to find the exact value of this sum. This is where it’s convinient to recognize that . Since we know what are Bernoulli numbers, we can use the following formula for Riemann zeta-function:

There are many ways of proving this formula, but none of them is elementary.

Recall that , so for we have . Hence completing the proof

**Remark:** One can directly caculate the value of as done by Euler while solving the Basel problem (though at that time the notion of convergence itself was not well defined):

The Pleasures of Pi, E and Other Interesting Numbers by Y E O Adrian [Copyright © 2006 by World Scientific Publishing Co. Pte. Ltd.]

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