数论讲义-黎曼Zeta函数.pdf

Riemann Zeta Function
Bent E. Petersen
January 23, 1996
Contents
Introduction........................ 1
TheZetaFunction.................... 2
TheFunctionalEquation................ 6
TheZerosoftheZetaFunction............. 7
StieltjesandHadamard................. 9
............. 11
Bibliography....................... 12
Introduction
The notes on the Riemann zeta function reproduced below are informal lecture
notes from two lectures from the plex variable course that I taught
20 years ago. Much of the material is cribbed from the books of Edwards and
Conway (see bibliography) and, of course, from Riemann’s 1859 paper on the
distribution of primes. The only original mathematics that I can claim is any
errors that I may have added. I have not updated the notes except to correct
errors. 1
These notes were prepared using LATEX2ε. The original notes, distributed in
February of 1976, were duplicated using hand–written ditto masters. We’ve
come a long way in desktop mathematical document preparation!
The purpose of these lectures on the zeta function was to illustrate some inter-
esting contour integral arguments in a nontrivial context and to make sure that
the students learned about the Riemann hypothesis – an important part of our
mathematical heritage and culture.
1Thanks to Mary Flahive for pointing out numerous errors.
1
2 Riemann Zeta Function
The Zeta Function
z ≥ >
If <e 1+ where 0then
Xn   Xn   Xn
z −−
 k−z  =  k−<e  ≤ k 1 (1)
k=m k=m k=m
P
∞−z
| n | { z ∈ C |<e z ≥ }
implies n=1 converges uniformly on 1+ .Thusthe
series
X∞
ζ(z)= n−z (2)
n=1
|<e z> }
converges normally in the half plane H = { z ∈ C 1 and so defines
an analytic function ζ in H. The function ζ is called the Riemann zeta function.
Note substituting nt for t yields
Z ∞ Z ∞
Γ(z)= e−t tz−1 dt = nz e−nt tz−1 dt. (3)
0 0
Therefore
X∞ Z ∞
ζ(z)Γ(z)= e−nt tz−1 dt (4)
n=1 0
z>
for <e 1. Now