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Riemann Zeta Function
The Riemann Zeta Function

The zeta function was first introduced by Leonhard Euler (17071708), who used it in the study of prime numbers.

In particular, Euler used its properties to show that
βπ prime
1π
diverges.

This shows in particular, that there are infinitely many primes, but also some information about their distribution.

Bernhard Riemann (18261866) used this function (a century after Euler) to obtain results on the asymptotic distribution of prime numbers.

And Riemann zeta function is usually along with the Riemann hypothesis to specify it's relation to prime numbers.
Introduction to the Zeta Function
Recall
ββπ=1
1π
diverges (harmonic series),
but
ββπ=1
1π^{π }
converges for all π >1.
That is
ββπ=1
1π^{π }
β€1+ββ«1
1π₯^{π }
ππ₯=1+11βπ
1π₯^{π β1}
βο½1
=1β11βπ
=π π β1
for (π >1)
Now consider π ββ instead of π ββ
DefinitionFor π ββ with Re π >1, the zeta function is defined as
π(π )=ββπ=1
1π^{π }
 It is traditional to call the complex variable "π " instead of "π§".
 What is π^{π } for π ββ? Note that for real π , we have that π^{π }=β―^{ln ππ }=β―^{π ln π}, so define
π^{π }=β―^{π log π}=β―^{π ln π} for π ββ
Convergence of π(π )
Does ββπ=1
1π^{π }
converge for Re π >1?
Since π^{π }=β―^{π ln π}, we have that π^{π }=β―^{π ln π}
=β―^{Re π ln π}=π^{Re π }. Thus
ββπ=1
1π^{π }
=ββπ=1
1π^{Re π }
,
and since Re π >1, the series on the right converges. Thus ββπ=1
1π^{π }
converges absolutely in {Re π >1}.
In fact, the convergence is uniform in {Re π β₯π} for any π>1, and this can be used to show that π(π ) is analytic in {Re π β₯1}
Analytic Continuation of the Zeta Function
One can now show the following (this theorem goes back to Riemann):
TheoremThe zeta function has an analytic continuation into β\{1}, and this continuation satisfies that π(π )ββ as π β1.
Slightly easier to construct is an extension to the right half plane {Re π >0}, minus the point 1, and we outline this construction here.
Motivation in β:
πβπ=1
1π^{π }
=π+1β«1
1π₯^{π }
ππ₯+πβπ=1
πΏ_{π}(π )
where
πΏ_{π}(π )=1π^{π }
βπ+1β«1
1π₯^{π }
ππ₯.
That is
πβπ=1
1π^{π }
βπ(π ) =π+1β«1
1π₯^{π }
ππ₯ β1/(π β1) +πβπ=1
πΏ_{π}(π ), where πΏ_{π}(π )=1π^{π }
βπ+1β«1
1π₯^{π }
ππ₯.
Observe that πβπ=1
πΏ_{π}(π ) is analytic in {Re π >0}. One can show that πβπ=1
πΏ_{π}(π ) converges, as πββ, to an analytic function π»(π ) in {Re π >0}. Thus
π(π )=1π β1
+π»(π ) holds for Re π >1,(*)
where
 π»(π ) is analytic in {
Re π >0}.
 π β¦
1π β1
is analytic in {Re π >0}\{1}
Therefore (*) can be used to define the zeta function in all of {Re π >0}\{1}.
This definition agrees with the original definition in {Re π >1}.
Riemann was actually able to extend the zeta function to an analytic function in all of β\{1}
The Zeros of the Zeta Function
Of much interest are the zeros of the zeta function, i.e. those π ββ, for which π(π )=0.
One can show:
TheoremThe only zeros of the zeta function outside of the strip {0β€Re π β€1} are at the negative even integers, β2, β4, β6,β―
 The zeros at β2, β4, β6,β― are often called the "trivial zeros", and the region to be studied remains the strip {0β€
Re π β€1}.
 A key result is that zeta has no zeros on the line {
Re π =1}, this is an essential fact in the proof of the prime number theorem.
 From the fact that zeta has no zeros on {
Re π =1}, it can easily be deduced that it has no zeros on {Re π =0} either, via a functional equation.
The Riemann Hypothesis
In Riemann's seminal paper in which he proved the analytic continuation of the zeta function to β\{1}, Riemann initiated important insights into the distribution of prime numbers. In this paper, Riemann expressed his belief in the veracity of the following:
Conjecture (Riemann Hypothesis)In the strip {0β€Re π β€1}, all zeros of π are on the line {Re π =12
}
Much research has been done in attempts to prove this conjecture:
 π(π ) has infinitely many zeros in {0<
Re π <1}. 
The asymptotic distribution of the zeros of π in {0<
Re π <1} is known. 
At least one third of the zeros in {0<
Re π <1} lie on the critical line {Re π =12
}. 
Trillions of zeros of zeta have been calculated  so far all of them lie on the critical line.

Numerical evidence and much research point to the validity of this conjecture, but it is to this day unproved and remains one of the most famous unsolved problems in mathematics.

The Riemann Hypothesis is on the list of seven "Millennium Prize Problems" (declared by the Clay Mathematics Institute in 2000). Only one of these has been solved so far (as of summer 2013)  the socalled Poincare Conjecture (by Grigori Perelman).

the riemann Hypothesis has strong implications on the distribution of prime numbers and on the growth of many other important arithmetic functions. It would greatly sharpen many numbertheoretic results.
The Power of Complex Analysis
Complex analysis is an extremely powerful field. This is demonstrated for example, by the ability to prove a deep theorem in number theory, the Prime Number Theorem, using complex analysis.
The Prime Counting Function
Let π(π₯)=number of primes less than or equal to π₯. This function is called the prime counting function. Example:
π(1)=0
π(2)=1
π(3)=2
π(4)=2
π(5)=3
π(6)=3
π(7)=π(8)=π(9)=π(10)=4
π(11)=π(12)=5
β―
It seems impossible to find an explicit formula for π(π₯). One thus studies the asymptotic behavior of π(π₯) as π₯ becomes very large.
Theorem (Prime Number Theorem)π(π₯)~π₯ln π₯
as π₯ββ.
Note: The symbol "~" means that the quotient of the two quantities approaches 1 as π₯ββ, i.e.
π(π₯)π₯/ln π₯
β1 as π₯ββ.
A Brief History of π(π₯)
 Euler (around 1740) discovered the connection between the zeta function π(π ) (for real values of π ) and the distribution of prime numbers.

60 years later, Legendre and Gauss conjectured the prime number theorem, after numerical calculations.

Another 60 years later, Tchebychev showed that there are constants π΄,π΅ (with 0<π΄<π΅) such that π΄
π₯ln π₯
β€π(π₯)β€π΅π₯ln π₯
. 
In 1859, Riemann published his seminal paper "On the Number of Primes Less Than a Given Magnitude". In this paper Riemann constructed the analytic continuation of the zeta function and introduced revolutionary ideas, connection its zeros to the distribution of prime numbers.

Hadamard and de la Vallee Poussin used these ideas and independently proved the Prime Number Theorem in 1896.

The main step in their proof is to establish that π(π ) has no zeros on {
Re π =1}.
How is π(π ) Related to Prime Number
Euler discovered:π(π )= βπ
11βπ^{βπ }
where the (infinite!) product is over all primes.
Proof
π(π )=11^{π }
+12^{π }
+13^{π }
+14^{π }
+15^{π }
+16^{π }
+17^{π }
+β―
=
1+12^{π }
+14^{π }
+β―1+13^{π }
+19^{π }
+β―1+15^{π }
+125^{π }
+β―1+17^{π }
+149^{π }
+β―β―
= βπ
ββπ=0
1π^{ππ }
= βπ
11β1π^{π }
The Riemann Zeta Function and Prime Numbers
π(π )= βπ
11β1π^{π }
Note:
 This product formula shows that π(π )β 0 for
Re π >1.  The key step in the proof of the prime number theorem is that π has no zeros on {
Re π =1}.  The details of the proof of the prime number theorem go beyond the introduction scrope.
 The prime number theorem says that π(π₯)~
π₯ln π₯
, but it doesn't have any information about the difference π(π₯)βπ₯ln π₯
.  However, the prime number theorem can also be written as π(π₯)~
Li(π₯), where Li(π₯)=π₯β«2
1ln π‘
ππ‘ is the (offset) logarithmic integral function
The Riemann Hypothesis and Prime Numbers
 The proofs of the prime number theorem by Hadamard and de la Vallee Poussin actually show that π(π₯)=
Li(π₯)+error term, where the error term grows to infinity at a controlled rate.  von Koch (in 1902) was able to give best possible bounds on the error term, assuming the Riemann hypothesis is true. Schoenfeld (in 1976) made this precise and proved that the Riemann hypothesis is equivalent to
π(π₯)βli(π₯)<π₯ln π₯8π
,
where li(π₯)=π₯β«0
1ln π‘
ππ‘ is the (unoffset) logarithmic integral function, related to Li(π₯) via Li(π₯)=li(π₯)βli(2).
the veracity of the Riemann Hypothesis would therefore imply further results about the distribution of prime numbers, in particular, they'd be distributed beautifully regtularly about there "expected" locations.
Final Remarks
 Complex numbers, algebra, geometry and topology in the complex plane, complex functions;
 complex dynamics, Julia sets of quadratic polynomials, the Mandelbrot set, conjecture of local connectedness of its boundary;
 complex differentiation, the CauchyRiemann equations, analytic ufnctions;
 Conformal mappings, inverse functions, Mobius transformations, the Riemann mapping theorem;
 Complex integration Cauchy theory and consequences (such as Liouville's theorem, the maximum principle, etc.)
 Complex series, power (Taylor) series, the Riemann zeta function and its relation to prime numbers, and the Riemann Hypothesis.