An introduction to the Riemann zeta function
Michel Waldschmidt
Content of this file (29 pages).
• p.2–3: a list of references.
• p. 4–22: from the bookArithmeticsbyM. Hindry: p. 127–131, 134–139, 148–155.
• p. 23–29: from El´ements d’analyse et d’alg`ebre (et de th´eorie des nom-´ bres) byP. Colmez : p. 40–43 and 51–53.
Further main references (not included in this file):
• Lectures on the Riemann zeta function byH. Iwaniec, Chap. 1–12, p.
3–43.
• An introduction to the theory of the Riemann zeta-functionbyS. J. Pat- terson, Chap. 1–11, p. 1–66.
• An introduction to zeta functionsbyP. Cartier, p 1–63.
1
References
LMFDB is the database of L-functions, modular forms, and related objects. It is an extensive database of mathematical objects arising in Number Theory.
http://www.lmfdb.org/
A basic course in number theory, containing the basic elements of analytic num- ber theory, is
• M. Hindry, Arithmetics, Universitext, Springer, London, 2011. Trans- lated from the 2008 French original.
Perhaps the simplest and more direct introduction to the Riemann zeta function is contained in the recent small (and a bit synthetic) book :
• H. Iwaniec,Lectures on the Riemann zeta function, vol. 62 of University Lecture Series, American Mathematical Society, Providence, RI, 2014.
Chapters 5–7 (p.21–26) present the theory from the beginning to the functional equation; chapters 8–11 (p.27–42) contain the rest of the basic theory, ending with the proof of PNT.
A discussion of the special values of the Riemann zeta function is included in
• P. Cartier, An introduction to zeta functions, in From number theory to physics (Les Houches, 1989), Springer, Berlin, 1992, pp. 1–63.
One of the most classical reference is
• G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, Oxford University Press, Oxford, sixth ed., 2008. Revised by D.
R. Heath-Brown and J. H. Silverman, With a foreword by Andrew Wiles.
Other classical books, all containing the basic theory of⇣(s), are:
• A. E. Ingham, The distribution of prime numbers, Cambridge Mathe- matical Library, Cambridge University Press, Cambridge, 1990. Reprint of the 1932 original, With a foreword by R. C. Vaughan.
2
• T. M. Apostol,Introduction to analytic number theory, Springer-Verlag, New York-Heidelberg, 1976. Undergraduate Texts in Mathematics.
• A. A. Karatsuba,Basic analytic number theory, Springer-Verlag, Berlin, 1993. Translated from the second (1983) Russian edition and with a preface by Melvyn B. Nathanson.
• S. J. Patterson, An introduction to the theory of the Riemann zeta- function, vol. 14 of Cambridge Studies in Advanced Mathematics, Cam- bridge University Press, Cambridge, 1988.
• H. Davenport,Multiplicative number theory, vol. 74 of Graduate Texts in Mathematics, Springer-Verlag, New York, third ed., 2000. Revised and with a preface by Hugh L. Montgomery.
• H. L. Montgomery and R. C. Vaughan,Multiplicative number theory.
I. Classical theory, vol. 97 of Cambridge Studies in Advanced Mathemat- ics, Cambridge University Press, Cambridge, 2007.
• G. Tenenbaum,Introduction to analytic and probabilistic number theory, vol. 163 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, third ed., 2015. Translated from the 2008 French edition by Patrick D. F. Ion.
• P. Colmez,El´ements d’analyse et d’alg`ebre (et de th´eorie des nombres),´ les ´editions de l’´ecole polytechnique, 2011, 678 pages.
3
§1. Elementary Statements and Estimates 127
1.6. Theorem. As x tends to infinity, we have the following asymptotic behavior:
π(x;a, b) := card{p prime , p!x, p ≡amodb}∼ x
φ(b) logx · (4.4) In this section, we will expand on some so-called “elementary” methods (which in this context means that they do not involve complex variables) which can be used to prove the previous assertions, except for Dirichlet’s theorem on arithmetic progressions and the prime number theorem. They will however allow us to prove a partial version: there exist two constants, c1, c2 >0, such that c1x/logx! π(x)!c2x/logx.
1.7. Lemma. The following estimate holds: nlog 2! log!2n
n
"
!nlog 4.
Proof. From the binomial theorem, we know that !2n
n
"
! #2n k=0
!2n
k
" = (1+1)2n = 4n. Next, we have the following lower bound:
$2n n
%
= (2n)!
(n!)2 = 2n(2n−1)· · ·(n+ 1)
n(n−1)· · ·1 "2n. #
1.8. Lemma. The following formula holds: ordp(n!) = #
m!1
&
n pm
'
; furthermore, the sum can be restricted to m!logn/logp.
Proof. Writen! =1·2·3· · ·n =(n
k=1k. The number of integers! nwhich are divisible by p is ⌊n/p⌋, and the number of integers ! n divisible by p2 is ⌊n/p2⌋, etc. Thus ordp(n!) is the sum of the ⌊n/pm⌋. Finally, pm ! n is equivalent to m! logn/logp, hence the first statement is proved. # We can therefore write
log$2n n
%
= )
p"2n
ordp
$2n n
%
logp= )
p"2n
⎛
⎝)
m!1
&
2n pm
'
−2&
n pm
'⎞
⎠logp.
(4.5) To find a lower bound, we only keep the terms that satisfy n < p! 2n. In fact, such a p clearly divides !2n
n
"
= (2n)!/(n!)2, and thus we obtain nlog 4 "log$2n
n
%" )
n<p"2n
logp= θ(2n)−θ(n).
From this, we obtain an upper bound of the form θ(x) !Cx. This is true because
θ(2m) =
m)−1 k=0
θ(2k+1)−θ(2k)!
m)−1 k=0
2klog 4 = (2m−1) log 4.
Therefore, if 2m !x < 2m+1, then
θ(x)!θ(2m+1)! 2m+1log 4! (2 log 4)x. (4.6) To obtain an upper bound, we could notice that ⌊2u⌋ −2⌊u⌋ always equals 0 or 1 and equals 0 whenever u <1/2. Thus
nlog 2! log$2n n
%
= )
p"2n
⎛
⎝)
m!1
&
2n pm
'
−2&
n pm
'⎞
⎠logp
! )
p"2n
$log(2n) logp
%
logp= log(2n)π(2n).
From this, we have a lower bound of the form π(x) " Cx/logx. This is true because if 2n! x <2(n+ 1), then
π(x) "π(2n)" nlog 2
log(2n) ". x
2 −1/ log 2
logx · (4.7) Furthermore, we can easily see that
θ(x) =)
p"x
logp !logx)
p"x
1 =π(x) logx. (4.8) Next, notice that for 2! y < x,
π(x)−π(y) = )
y<p"x
1! 1 logy
)
y<p"x
logp= 1
logy (θ(x)−θ(y)). It follows that
π(x)! θ(x)
logy +π(y) ! θ(x) logy +y.
By choosing y = x/(logx)2 and by recalling the previous inequality (4.8), we have
θ(x)
logx ! π(x) ! θ(x)
logx+ 2 log logx + x
(logx)2 · (4.9) To summarize, it is easy to see from inequalities (4.6), (4.7), (4.8) and (4.9) that (θ(x)∼ x) is equivalent to (π(x)∼x/logx) and that
C1x!θ(x)! C2x and C3x/logx!π(x)! C4/logx. (4.10) Furthermore, the following comparison of the functionθ(x) to the function ψ(x) is not difficult to see:
§1. Elementary Statements and Estimates 129
θ(x)!ψ(x) := )
pm"x
logp= θ(x) +θ(√x) +θ(√3x) +. . .
!θ(x) + log(x)
log 2 θ(√x)!θ(x) +Clogx√ x.
Finally, if we denote bypnthenth prime number, we haveπ(pn) =nby def- inition. The prime number theorem therefore implies that n∼ pn/log(pn) and that pn ∼ nlogn. We can check that the latter statement is in fact equivalent to the prime number theorem.
1.9. Lemma. (Abel’s formula) Let A(x) :=#
n"xan and f be a function
of class C1. Then, )
y<n"x
anf(n) =A(x)f(x)−A(y)f(y)− 0 x
y
A(t)f′(t)dt. (4.11)
Proof. We first point out that 1n+1
n A(t)f′(t)dt = A(n)1n+1
n f′(t)dt = A(n) (f(n+ 1)−f(n)). Therefore, setting N =⌊x⌋ and M = ⌊y⌋ yields
0 N M
A(t)f′(t)dt =
N)−1 n=M
0 n+1 n
A(t)f′(t)dt=
N)−1 n=M
A(n+ 1) (f(n)−f(n))
= )N n=M+1
f(n)(A(n−1)−A(n)) +f(N)A(N)−A(M)f(M)
=− )N n=M+1
f(n)an +f(N)A(N)−A(M)f(M).
This proves the formula whenxandy are integers. For the general formula, observe that
0 x
⌊x⌋
A(t)f′(t)dt= A(⌊x⌋) (f(x)−f(⌊x⌋)) =A(x)f(x)−A(⌊x⌋)f(⌊x⌋).#
Applications. 1) The formula gives a fairly precise comparison between the “sum” and the “integral” (see Exercise4-6.10 for some refinements). To be more precise, if we take an = 1 and integrate by parts, we have:
)N n=M+1
f(n) =0 N M
f(t)dt+0 N M
(t− ⌊t⌋)f′(t)dt. (4.12) If we choose f(t) = 1/t, we obtain
)N n=1
n1 = 1 +0 N 1
dt t −
0 N 1
(t− ⌊t⌋) dt t2
= logN +$ 1−
0 ∞
1
(t− ⌊t⌋) dt t2
%
+0 ∞
N
(t− ⌊t⌋) dt t2
= logN +γ+O. 1 N
/, where γ := 1−1∞
1 (t− ⌊t⌋) dt
t2 is Euler’s constant.
2) Take an = 1, so A(t) =⌊t⌋, y= 1 and f(t) = logt. We therefore have log (⌊x⌋!) =⌊x⌋log(x)−
0 x 1
⌊t⌋dt t
= xlogx− 0 x
1
dt+ (⌊x⌋ −x) logx− 0 x
1
⌊t⌋ −t t dt
= xlogx−x+O(logx).
We should point out that Stirling’s formula gives a slightly more precise statement, namely n! ∼ nne−n√
2πn, and hence log(n!) = nlogn−n + 12 logn+ 12 log(2π) +ϵ(n) where limn→∞ϵ(n) = 0.
Furthermore, we see that log (⌊x⌋!) = )
p"x
ordp(⌊x⌋!) logp
= )
p"x
)
m!1
&
x pm
' logp
=x)
p"x
logp
p +)
p"x
logp.2 x p
3− x p
/+)
p"x
)
m!2
&
x pm
' logp
=x)
p"x
logp
p +O(x),
where the last estimate comes from the upper bound θ(x) =#
p"xlogp =
O(x), from (4.6) and the estimate )
p"x
)
m!2
&
x pm
'
logp !x)
p"x
)
m!2
logp
pm = x)
p"x
logp
p(p−1) = O(x).
From this, we can deduce the first formula in Proposition 4-1.2, )
p"x
logp
p = logx+O(1). (4.13)
§2. Holomorphic Functions (Summary/Reminders) 131
To get the second, we apply Abel’s formula (Lemma 4-1.9) letting f(t) = 1/logtandan = logp/pifn= pis prime andan = 0otherwise. By setting A(x) =#
p"x
logp
p , we have that )
p"x
1p = )
n"x
anf(n)
= A(x)
logx +0 x 2
A(t)dt t(logt)2
= 1 +O(1/logx) +0 x 2
dt
tlogt +0 x 2
(A(t)−logt) t(logt)2 dt
= log logx−log log 2 + 1 +0 ∞ 2
(A(t)−logt)
t(logt)2 dt+O(1/logx).
2. Holomorphic Functions (Summary/Reminders)
This section, without proofs, is a summary of some of the fundamental properties of functions of a complex variable that we will be using. It could be helpful to use [74] as a reference.
Concerning series, we will use the product rule for calculating the product of two absolutely convergent series:
4 ∞ )
n=0
an
5 4 ∞ )
n=0
bn 5
= )∞ n=0
4 n )
k=0
akbn−k 5
,
as well as rearrangement of the order of summation in a series withpositive terms am,n:
)∞ n=0
4 ∞ )
m=0
am,n 5
= )∞
m=0
4 ∞ )
n=0
am,n 5
. A power series S(z) = #∞
n=0anzn is said to have a radius of convergence R " 0 (possibly R = 0 or R = +∞) if the series converges for all |z| < R and diverges for all |z| > R; furthermore, the convergence is absolute in the interior of the disc of convergence and the function is of classC∞ with S(k)(z) =#∞
n=kn(n−1)· · ·(n−k+ 1)anzn−k. In fact, the functionS can be expanded as a power series around every point z0 ∈ D(0, R), in other words, for everyz ∈D(z0, r)⊂ D(0, R), we have S(z) =#∞
n=0bn(z−z0)n (with bn = S(n)(z0)/n!). Such a function only has a finite number of zeros in every closed disc (or compact set) which is contained in D(0, R).
We define the multiplicity of a zero, z0, as the integer k such that S(z) =
The series converges normally at every point in the open disk D(1,1) = {z ∈ C | |1−z| < 1}, and the convergence is uniform (and also normal) on every closed disk with center 1 and radius r < 1. Therefore, F(z) is holomorphic on D(1,1). If z is a real number in the interval ]0,2[, we can see that F(z) = logz (ordinary logarithm), and in particular,
exp (F(z)) =z.
The previous formula indicates that the two functions, the identity and exp◦F, which are analytic on the disk D(1,1), coincide on the segment ]0,2[ and hence on the whole disk. Thus F defines a complex logarithm on the disk |z−1| <1.
2.8. Definition. Let f(z) be a holomorphic function on U. We say that F(z) is a branch of the logarithm of f on U (and we write, with a slight abuse of notation, F(z) = logf(z)) if F(z) is holomorphic and if exp (F(z)) =f(z).
2.9. Remark. If F(z) is a branch of the logarithm of f, then f is never 0 on U, we have |exp (F(z))|= exp (ReF(z)) =|f(z)|, and hence
Re logf(z) = log|f(z)|.
Likewise, f′(z)/f(z) =F′(z) exp (F(z))/exp (F(z)) =F′(z), and also d
dz logf(z) = f′(z) f(z) ·
Finally, ifF1 andF2 are two logarithms, then F2(z) =F1(z) + 2kπi on any connected set U.
This remark suggests that we should construct the logarithm of f(z)as an antiderivative f′(z)/f(z), with the condition that f is not zero. We have seen that this is possible if U is simply connected.
2.10. Proposition. Let U be a simply connected open subset of the complex plane and f(s) a holomorphic function without any zeros in U.
Then there exists a holomorphic branch F(s) = logf(s) on U. Two such branches differ by an integer multiple of 2πi.
We will finish this summary by explaining the notion of an infinite prod- uct. The first idea consists of saying that a product is convergent if limN!N
n=0pn exists. This could be confusing because it is not true that such a product is zero if and only if one of the factors is zero. For example, it can be easily checked that
§3. Dirichlet Series and the Function ζ(s) 135
N→∞lim
"N n=1
#1− 1 n+ 1
$= 0.
We can overcome this inconvenience by defining infinite products a little differently. Observe first that a necessary condition for the convergence of a non-zero product !
npn is to have limnpn = 1; it therefore does not hurt to assume that pn = 1 +un where un tends to zero. In particular, log(1 +un) = %∞
k=1(−1)kukn/k is well-defined when |un| < 1 and hence when n! n0, which justifies the following definition.
2.11. Definition. A product !∞
n=0(1+un) isconvergent(resp. absolutely convergent) if there exists n0 such that |un| < 1 for all n ! n0 and the series %∞
n=n0log(1 +un) is convergent (resp. absolutely convergent). A product of functions!∞
n=0(1+un(z)) isuniformly convergentonK if there exists n0 such that |un(z)| < 1 for all n ! n0 and z ∈ K and the series
%∞
n=n0log(1 +un(z)) is uniformly convergent (on K).
2.12. Lemma. A product P := !∞
n=0(1 +un) is absolutely convergent if and only if the series %∞
n=0|un| is convergent. If P is convergent, it is zero if and only if one of the factors 1 +un is zero.
2.13. Proposition. Let (un(z)) be a sequence of holomorphic functions on an open setU such that the series%
nlog(1+un(z)) converges uniformly on every compact subset of U.
i) Then the function defined by the infinite product P(z) := "∞
n=0
(1 +un(z)) is holomorphic on U.
ii) For every z0 ∈ U, only a finite number of pn(z) := 1 +un(z) are zero at z0, and hence
ordz0P(z) = &∞
n=0
ordz0pn(z).
3. Dirichlet Series and the Function ζ(s)
We call a Dirichlet series a series of the form F(s) = %∞
n=1
an
ns · We will now state its first important property.
3.1. Proposition. Let F(s) = %∞ n=1
an
ns be a Dirichlet series that we will assume to be convergent at s0. Then it converges uniformly on the sets EC,s0 ={s∈ C| Re(s−s0)!0,|s−s0| "CRe(s−s0)}.
Proof. For M ! 1, set AM(x) := %
M <n!xann−s0. By the hypothesis, we then have |AM(x)| " ϵ(M) where ϵ(M) tends to zero as M tends to infinity. Abel’s formula gives
&
M <n!N
ann−s = &
M <n!N
ann−s0n−(s−s0)
= AM(N)N−(s−s0)+ (s−s0)' N M
AM(t)t−(s−s0+1)dt.
We can find an upper bound for the integral as follows:
(( (( (
' N M
AM(t)t−(s−s0+1)dt (( ((
(" ϵ(M)' N M
t−(σ−σ0+1)dt
= ϵ(M) M−(σ−σ0)−N−(σ−σ0) (σ−σ0) ·
By restricting to an angular sector EC,s0 bounded by σ− σ0 = Re(s) − Re(s0)!0 and |s−s0|= C(σ−σ0), we obtain
(( (( ((
&
M <n!N
ann−s (( ((
(( "ϵ(M)(1 +C),
which suffices to show the uniform convergence on this sector (cf. the figure
below). #
the domain |s−s0|" σ−σ0 cosθ
The following corollary is a result of the general theorems recalled in the previous section (in particular, Proposition 4-2.6).
3.2. Corollary. Every Dirichlet series F(s) =%∞ n=1
an
ns has an abscissa of convergence, say σ0, such that the series converges for Re(s) > σ0 and
§3. Dirichlet Series and the Function ζ(s) 137
diverges for Re(s) < σ0. Furthermore, the function F defined by the se- ries is holomorphic in the half-plane of convergence Re(s) > σ0, and its derivatives are given by F(k)(s) =%∞
n=1an(−logn)kn−s.
Proof. It suffices to let σ0 = inf{σ ∈ R | the series converges at σ}, then to observe that every compact set in the (open) half-plane of convergence is contained in a sector, as above, where the convergence is uniform. # 3.3. Remarks. 1) If a Dirichlet series converges at s0 = σ0 + it0 to the number S, then the proof of Proposition 4-3.1 (above) shows that S = limϵ→0+F(σ0 +ϵ+it0).
2) Set A(t) := %
n!tan. The previous proof allows us to establish the formula
&∞ n=1
ann−s =s ' ∞
1
⎛
⎝&
n!t
an
⎞
⎠t−s−1dt =s ' ∞
1
A(t)t−s−1dt. (4.14) In particular, if A(t) = %
n!tan is bounded, then the series converges whenever Re(s) >0.
The most famous Dirichlet series is the Riemann zeta function, defined by the series %∞
n=1n−s. It is well-known, at least for real values and the general case follows from the real case, that the abscissa of convergence is +1.
3.4. Theorem. (Euler Product) If Re(s) >1, then the following formula holds
ζ(s) =
&∞ n=1
1
ns = "
p
- 1− 1
ps .−1
. (4.15)
Proof. Notice that %
p|p−s| = %
pp−σ " %
nn−σ, hence the product is absolutely convergent. If Re(s) > 0, the convergence of geometric series allows us to write(1−p−s)−1 = %∞
m=0p−ms. By taking the product over the prime numbersp1, . . . , pr which are" T, we obtain
"
p!T
(1−p−s)−1 = "
p!T
/ ∞
&
m=0
p−ms 0
= &
m1, . . . , mr ! 1 p1, . . . , pr " T
(pm1 1· · ·pmr r)−s
= &
n∈N(T)
n−s,
where we denote byN (T)the set of integers all of whose prime factors are
"T. Thus whenever Re(s) >1, we have ((
(( ((
&∞ n=1
1
ns − "
p!T
- 1− 1
ps .−1((((
((= (( (( ((
&
n /∈N(T)
1 ns
(( ((
((" &
n>T
(( ( 1
ns ((
( = &
n>T
1 nσ · The last sum is the tail-end of a real convergent series (when σ:= Re(s) >
1) and thus tends to zero, which proves both the convergence of the product
and Euler’s formula. #
3.5. Corollary. The function ζ(s) does not have any zeros in the open half-plane Re(s) > 1. A holomorphic branch of logζ(s) for Re(s) > 1 can be constructed by setting
logζ(s) =&
p
&
m"1
p−ms
m · (4.16)
Furthermore, if we define the von Mangoldt function by Λ(n) =
1logp if n =pm,
0 if not, then
− ζ′(s)
ζ(s) = &
p
&
m"1
logp
pms = &∞
n=1
Λ(n)
ns · (4.17)
Proof. We know1−p−s ̸= 0 and that the product is convergent. The first assertion is therefore obvious. The second formula can be deduced from Euler’s formula by taking the series expansion of the logarithm (valid for
|x| <1) and summing:
log2(1−x)−13 = &∞
m=1
xm m ·
The second formula is therefore gotten by differentiating the first. # Interlude (I).These formulas can be generalized by replacingZandQby OK and K, and the uniqueness of the decomposition into prime factors by the uniqueness of the decomposition into prime ideals (Theorem 3-4.18).
We denote by IK the set of non-zero ideals and PK the set of non-zero (maximal) prime ideals in OK. Now we can introduce the Dedekind zeta function and prove that
ζK(s) := &
I∈IK
N(I)−s = "
p∈PK
21−N(p)−s3−1
for Re(s)> 1.
§4. Characters and Dirichlet’s Theorem 139
3.6. Proposition. The function ζ(s) can be analytically continued to a meromorphic function to the half-plane Re(s) > 0, with a unique pole at s= 1 with residue equal to +1.
Proof. The statement says that ζ(s) − 1/(s − 1), originally defined for Re(s) >1, has a holomorphic continuation to the half-plane Re(s)> 0. To prove this, we can write ζ(s), using the expression ⌊t⌋= %
n!t1, as
ζ(s) =&∞
n=1
1 ns =s
' ∞
1 ⌊t⌋t−s−1dt
=s ' ∞
1
t−sdt+s ' ∞
1
(⌊t⌋ −t)t−s−1dt
= 1
s−1 + 1 +s ' ∞
1
(⌊t⌋ −t)t−s−1dt.
We know that |⌊t⌋ − t| " 1. The last integral is hence convergent and defines a holomorphic function for Re(s)> 0. # 3.7. Remark. Actually, the function ζ(s)−1/(s−1) can be extended to the whole complex plane, and moreover,ζ(s)satisfies a functional equation (see Theorem 4-5.6 further down).
4. Characters and Dirichlet’s Theorem
4.1. Definition. If G is a finite abelian group, a homomorphism from G to C∗ is called a character. The set of characters of G forms a group denoted by G.ˆ
4.2. Proposition. The group Gˆ is isomorphic to the group G (but not canonically).
Proof. If G= Z/nZ, a character satisfies χ(1)∈ µn (where µn denotes as usual the group of nth roots of unity), and the map χ '→ χ(1) provides an isomorphism between Gˆ and µn. The latter is isomorphic to Z/nZ and hence to G. We will now show that G!1 ×G2 ∼= ˆG1 × Gˆ2; to see why this is true, a character χ of G1 ×G2 can be written as χ(g1, g2) = χ(g1, e2)χ(e1, g2), and, by settingχ1 =χ(·, e2)andχ2 =χ(e1,·), we obtain an isomorphismχ'→ (χ1,χ2) from G!1 ×G2 to Gˆ1 ×Gˆ2. The general case is now easy: we haveG∼= Z/n1Z×· · ·×Z/nrZ, hence
Gˆ ∼= (Z/n1Z×· · ·×Z/nrZ)b
∼= Z/n!1Z×· · ·×Z/n!rZ∼=Z/n1Z×· · ·×Z/nrZ∼= G. #
