Section 24.2 The Riemann Zeta Function
Subsection 24.2.1 A fundamental function
The most important such infinite process is the following fundamental function. It is one of the most studied, yet most mysterious functions in all of mathematics.Definition 24.2.1. Riemann zeta function.
We define the zeta function (denoted
as a function of
For now we'll keep the domain of

plot(zeta,0,4,ymin=-1,ymax=10)
)Historical remark 24.2.3. Bernhard Riemann.
Riemann, the quietly devout son of a Lutheran pastor, made ground-breaking contributions in nearly every area of mathematics. He did it in analysis (Riemann sums), in geometry (Riemannian metrics, later used by Einstein), in function theory (Riemann surfaces) β and in one paper that changed the course of number theory. He died quite young (around 40), but unlike some of his contemporaries had achieved wide recognition in his own lifetime for his advances.
Subsection 24.2.2 Motivating the Zeta function
The motivation for this definition comes from this function with the casexxxxxxxxxx
def _(n=[30,20,18,24,12,16]):
str = '$$'+' + '.join([r'\frac{1}{%s}'%d for d in divisors(n)])+'=%s$$'%sum([1/d for d in divisors(n)])
str2 = '$$' + ''.join([r'\left('+'+'.join([r'\frac{1}{%s^{%s}}'%(p, k) for k in [0..e]])+r'\right)' for (p,e) in factor(n)]) + '=%s$$'%prod([sum([p^(-k) for k in [0..e]]) for (p,e) in factor(n)])
pretty_print(html(str))
pretty_print(html("compare to "+str2))
xxxxxxxxxx
def _(e=(1,[0..6]),f=(2,[0..6])):
n = 2^e*3^f
pretty_print(html("You picked $%s=2^{%s}3^{%s}$"%(n,e,f)))
str = '$$'+' + '.join([r'\frac{1}{%s}'%d for d in divisors(n)])+'=%s$$'%sum([1/d for d in divisors(n)])
str2 = '$$' + ''.join([r'\left('+' + '.join([r'\frac{1}{%s^{%s}}'%(p,k) for k in [0..e]])+r'\right)' for (p,e) in factor(n)]) + '=%s$$'%prod([sum([p^(-k) for k in [0..e]]) for (p,e) in factor(n)])
pretty_print(html(str))
pretty_print(html("compare to "+str2))
Subsection 24.2.3 Being careful
So much for Euler's contribution, a very impressive one. The only problem with all this is that both of these things clearly diverge! Thus we cannot use a simple equality (Proposition 24.2.4. Integral test for series convergence.
Assume
Fact 24.2.5.
The infinite sum