In this last series the power of the terms is 2. In the harmonic series the
power of the denominators is silently equal to 1 and this value is critical. If the
power increases by a minuscule amount to a number just above 1 the series
converges, but if the power decreases by a minuscule amount to a value just
below 1, the series diverges. The harmonic series sits on the boundary between
convergence and divergence.
The Riemann zeta function
The celebrated Riemann zeta function ζ(s) was actually known to Euler in the
18th century but Bernhard Riemann recognized its full importance. The ζ is the
Greek letter zeta, while the function is written as:
Various values of the zeta function have been computed, most prominently,
ξ(1) = ∞ because ξ(1) is the harmonic series. The value of ξ(2) is π^2 /6, the result
discovered by Euler. It has been shown that the values of ξ(s) all involve π when
s is an even number while the theory of ξ(s) for odd values of s is far more
difficult. Roger Apéry proved the important result that ξ(3) is an irrational
number but his method did not extend to ξ(5), ξ(7), ξ(9), and so on.
The Riemann hypothesis
The variable s in the Riemann zeta function represents a real variable but this
can be extended to represent a complex number (see page 32). This enables the
powerful techniques of complex analysis to be applied to it.