The zeta function is closely related to prime numbers. For example, the aforementioned fact that there are infinitely many primes can also be seen using the zeta function: if there were only finitely many primes then ζ(1) would have a finite value. However, the harmonic series 1 + 1/2 + 1/3 + 1/4 + ... diverges (i.e., exceeds any given number), so there must be infinitely many primes. Another example of the richness of the zeta function and a glimpse of modern algebraic number theory is the following identity (Basel problem), due to Euler,
What does it mean when a harmonic series diverges?
exceeds any given number exceeds any given number exceeds any given number exceeds any given number
diverges (i.e., exceeds any given number), so there must be infinitely many primes.
diverges (i.e., exceeds any given number), so there must be infinitely many primes.
What does it mean when a harmonic series diverges?