nyuuzyou/wikis huggingface dataset:
Riemann Hypothesis/Historical Note

== Historical Note on Riemann Hypothesis ==
The Riemann Hypothesis was stated by in his $1859$ article .
It is the last remaining statement which has not been resolved is the Riemann Hypothesis.
This problem is the first part of no. $8$ in the Hilbert 23, and also one of the Millennium Problems, the only one to be in both lists.
It is important in work on the distribution of primes.
In 's words, in his posthumous papers:
:[ These theorems ] follow from an expression for the function $\map \zeta s$ which I have not simplified enough to publish.
As put it:
:We still have not the slightest idea of what that expression should be. ... In general, Riemann's intuition is highly geometrical; but this is not the case for his memoir on prime numbers, the one in which that intuition is the most powerful and mysterious.
In $1914$, proved the Critical Line Theorem, that there exist an infinite number of nontrivial zeroes of the Riemann $\zeta$ function on the critical line.
This was again demonstrated in $1921$, by together with .
In $1974$, demonstrated that At Least One Third of Zeros of Riemann Zeta Function on Critical Line.
By $1983$, systematic exploration of the critical strip with the aid of computers had shown that the first $3 \, 500 \, 000$ nontrivial zeroes were all located on the critical line.
While this is compelling, it is far from being a proof.
In December $1984$, it was announced that had found a proof, but this was shown to be flawed.
== Sources ==
* : $0,5$
* : $0 \cdotp 5$
* : Chapter $\text {A}.32$: Riemann ()
* : $0 \cdotp 5$
* 
* 
Category:Historical Notes
Riemann Hypothesis/Zeroes/1

== Zero of Riemann Zeta Function ==
The first zero of the Riemann $\zeta$ function is positioned at:
:$\dfrac 1 2 + i \paren {14 \cdotp 13472 \, 5 \ldots}$
== Sources ==
* : $14 \cdotp 13472 \, 5 \, \ldots$
Category:Riemann Hypothesis
Riemann Hypothesis implies Odd Number is Sum of at most 5 Primes/Historical Note

== Historical Note on Riemann Hypothesis implies Odd Number is Sum of at most 5 Primes ==
When this result was cited by in the $1997$ edition of , this was the limit of knowledge.
However, has since come up with a proof of this result which does not require the Riemann Hypothesis.
Category:Historical Notes
Riemann Integrable Dirac Function does not Exist

== Theorem ==
Let $\delta : \R \to \R$ be a real function.
Let $\phi : \R \to \R$ be a smooth function vanishing outside $\closedint a b$.
Let $a \in \R_{> 0}$ be a real number.
Suppose $\delta$ is Riemann integrable on $\closedint {-a} a$.
Suppose for every $\phi$ we have that:
:$\ds \int_{-a}^a \map \delta x \map \phi x = \map \phi 0$
Then $\delta$ does not exist.
== Proof ==
 $\delta$ exists.
Let $\phi$ be a test function of the following form:
:$\map \phi x = \begin {cases} \map \exp {\dfrac 1 {x^2 - 1} } & : \size x < 1 \\ 0 & : \size x \ge 1 \end {cases}$
Let $n \in \N$ be a natural number.
Let $\phi_n : \R \to \R$ be a real function such that:
:$\forall n \in N : \forall x \in \R : \map {\phi_n} x := \map \phi {n x}$
Then $\map \phi {nx}$ is smooth, vanishes outside $\ds \closedint {- \frac 1 n} {\frac 1 n}$ and:
:$\forall x \in \R : 0 \le \map \phi {nx} \le 1$
Hence:
This is a contradiction.
== Sources ==
* : Chapter $\S 6.1$: A glimpse of distribution theory. Test functions, distributions, and examples
Category:Dirac Delta Function
Riemann Integral Operator is Continuous Linear Transformation

== Theorem ==
Let $\struct {C \closedint a b, \norm {\, \cdot \,}_\infty}$ be the normed vector space of real-valued functions continuous on $\closedint a b \subseteq \R$ equipped with the supremum norm.
Let $T : C \closedint a b \to \R$ be the Riemann integral operator:
:$\ds \forall \mathbf x \in C \closedint a b : \map T {\mathbf x} = \int_a^b \map {\mathbf x} t \rd t$
Then $T$ is a continuous mapping.
== Proof ==
We have that Integral Operator is Linear.
Furthermore:
By Continuity of Linear Transformation between Normed Vector Spaces, $T$ is continuous.
== Sources ==
* : Chapter $\S 2.3$: The normed space $\map {CL} {X,Y}$
Category:Operator Theory
Category:Definite Integrals
Category:Linear Transformations
Category:Continuous Transformations
Riemann Integral Operator is Linear Mapping

== Theorem ==
Let $C \closedint a b$ be the space of continuous Riemann integrable functions.
Let $\R$ be the set of real numbers.
Let $I : C \closedint a b \to \R$ be the Riemann integral operator.
Then $I$ is a linear mapping.
== Proof ==
Let $x, y \in C \closedint a b$ be Riemann integrable.
Let $\alpha \in \R$.
=== Distributivity ===
=== Positive homogenity ===
== Sources ==
* : Chapter $\S 2.1$: Continuous and linear maps. Linear transformations
Category:Linear Transformations
Category:Operator Theory
Riemann P-symbol in terms of Gaussian Hypergeometric Function

== Theorem ==
Let:
:$\map f z = \operatorname P \set {\begin{matrix} a & b & c \\ \alpha & \beta & \gamma & z \\ \alpha' & \beta' & \gamma' \end{matrix} }$
where:
:$\operatorname P$ is the Riemann $P$-symbol
:$\alpha + \beta + \gamma + \alpha' + \beta' + \gamma' = 1$ 
:$\alpha - \alpha'$ is not a negative integer. 
Then: 
:$\ds \map f z = \paren {\dfrac {z - a} {z - b} }^\alpha \paren {\dfrac {z - c} {z - b} }^\gamma \map F {\alpha + \beta + \gamma, \alpha + \beta' + \gamma; 1 + \alpha - \alpha'; \dfrac {\paren {z - a} \paren {c - b} } {\paren {z - b} \paren {c - a} } }$
where $F$ is the Gaussian hypergeometric function.
== Proof == 
== Sources ==
* : $14.3$: Solutions of Riemann's P-equation by hypergeometric functions
Category:Differential Equations
Category:Riemann P-symbol
Category:Gaussian Hypergeometric Function
Riemann Removable Singularities Theorem

== Theorem ==
Let $U \subset \C$ be a domain.
Let $z_0 \in U$.
Let $f: U \setminus \set {z_0} \to \C$ be holomorphic.
Then the following are equivalent:
:$(1): \quad f$ extends to a holomorphic function $f: U \to \C$.
:$(2): \quad f$ extends to a continuous function $f: U \to \C$.
:$(3): \quad f$ is bounded in a deleted neighborhood of $z_0$.
:$(4): \quad \map f z = \map o {\dfrac 1 {\cmod {z - z_0} } }$ as $z \to z_0$.
== Proof ==
, we may assume that:
:$U = \Bbb D := \set {z \in \C: \cmod z < 1}$
and that $z_0 = 0$. 
Otherwise, restrict $f$ to a suitable disk centered at $z_0$, and precompose with a suitable affine map.
A holomorphic function is continuous.
Continuous Function is Locally Bounded.
So $(1) \implies (2) \implies (3)$.
Also, $(3) \implies (4)$ by definition. 
That $(2) \implies (1)$ follows from the proof of Cauchy's Residue Theorem.
(For completeness, we sketch a self-contained argument below.)
Now assume that $(4)$ holds.
That is:
:$z \cdot \map f z \to 0$ as $z \to 0$.
We need to show that $(2)$ holds.
By assumption, the function:
:$\map g z := \begin{cases} z \cdot \map f z & : z \ne 0 \\ 0 & : \text{otherwise} \end{cases}$
is continuous in $\Bbb D$ and holomorphic in $\Bbb D \setminus \set 0$.
By applying the direction $(2) \implies (1)$ to this function, we see that $g$ is holomorphic in $\Bbb D$.
We have:
:$\ds \map {g'} 0 = \lim_{z \mathop \to 0} \frac {\map g z} z = \lim_{z \mathop \to 0} \map f z$
so $f$ extends continuously to $\Bbb D$, as claimed.
To prove $(2) \implies (1)$, we use Morera's Theorem and Cauchy's Residue Theorem.
By Morera's Theorem, we need to show that:
:$\ds \int_C \map f z \rd z = 0$
for every closed curve $C$ in $\Bbb D$.
It follows from the Cauchy Integral Theorem that we only need to check that:
:$\ds \int_C \map f z \rd z = 0$
for a simple closed loop surrounding $0$, and that this integral is independent of the loop $C$.
Letting $C = \map {C_\epsilon} 0$ be the circle of radius $\epsilon$ around $0$, we see that:
:$\ds \size {\int_{C_\epsilon} \map f z \rd z} \le 2 \pi \epsilon \max_{z \mathop \in C_\epsilon} \size {\map f z} \to 0$
as $\epsilon \to 0$ (because $f$ is continuous in $0$ by assumption).
This completes the proof.
Category:Complex Analysis
Category:Continuous Functions
Riemann Surface/Examples/Torus

== Example of Riemann Surface ==
Let $f: \C \to \C$ be the complex function defined as:
:$\forall z \in \C: \map f z = \sqrt {z^2 + z + 1}$
The Riemann surface of $f$ is homeomorphic to the torus.
== Sources ==
* 
* 
Category:Examples of Riemann Surfaces
Riemann Zeta Function Converges where Real Part Greater than 1

== Theorem ==
Let $\zeta$ denote the Riemann zeta function:
:$\forall s = \sigma + i t \in \C: \ds \map \zeta s = \sum_{n \mathop = 1}^\infty \frac 1 {n^s}$
Then $\map \zeta s$ is convergent when $\sigma > 1$.
== Proof ==
== Sources ==
* 
* 
Category:Riemann Zeta Function
Riemann Zeta Function and Prime Counting Function

== Theorem ==
For $\map \Re s > 1$:
:$\ds \log \map \zeta s = s \int_0^{\mathop \to \infty} \frac {\map \pi x} {x \paren {x^s - 1} } \rd x$
where:
:$\zeta$ denotes the Riemann Zeta Function
:$\pi$ denotes the Prime-Counting Function.
== Proof ==
From the definition of the Riemann Zeta Function: 
By Derivative of Logarithm Function and the Chain Rule for Derivatives:
Hence:
Category:Riemann Zeta Function
Category:Number Theory
Riemann Zeta Function as a Multiple Integral

== Theorem ==
For $n \in \Z_{> 0}$, the Riemann zeta function is given by:
:$\ds \map \zeta n = \int_{\closedint 0 1^n} \frac 1 {1 - \prod_{i \mathop = 1}^n x_i} \prod_{i \mathop = 1}^n \rd x_i$
where $\closedint 0 1^n$ denotes the Cartesian $n$th power of the closed real interval $\closedint 0 1$.
== Proof ==
Category:Riemann Zeta Function
Category:Analytic Number Theory
Riemann Zeta Function at Even Integers

== Theorem ==
The Riemann $\zeta$ function can be calculated for even integers as follows:
where:
:$B_n$ are the Bernoulli numbers
:$n$ is a positive integer.
=== Corollary ===
==Proof 1==
==Proof 2==
== Also presented as ==
== Examples ==
== Also see ==
* Basel Problem
* Riemann Zeta Function at Odd Integers: still unsolved
== Historical Note ==
== Sources ==
* : $\S 19$: Series involving Reciprocals of Powers of Positive Integers: $19.35$
* : $\S 35$: Riemann Zeta Function $\map \zeta x = \dfrac 1 {1^x} + \dfrac 1 {2^x} + \dfrac 1 {3^x} + \cdots$: $35.26$
* : $0,1666666666666 \ldots$
* : $2$
* 
* : Chapter $\text {B}.20$: The Bernoulli Numbers and some Wonderful Discoveries of Euler
* : $\S 1.2.7$: Harmonic Numbers: $(6)$
* : $2$
* : $\S 21$: Series of Constants: Series Involving Reciprocals of Powers of Positive Integers: $21.35.$
Category:Riemann Zeta Function at Even Integers
Category:Examples of Riemann Zeta Function
Category:Bernoulli Numbers
Category:Analytic Number Theory
Riemann Zeta Function at Even Integers/Also presented as

== Riemann Zeta Function at Even Integers: Also presented as == 
This can also be seen rendered in the elegant form:
:$\map \zeta r = \dfrac 1 2 \size {B_r} \dfrac {\paren {2 \pi}^r} {r!}$
for $r = 2 n$, $n \ge 1$.
It can also be expressed using the archaic form of the Bernoulli numbers as:
== Sources ==
* : $\S 19$: Series involving Reciprocals of Powers of Positive Integers: $19.35$
* : $\S 35$: Riemann Zeta Function $\map \zeta x = \dfrac 1 {1^x} + \dfrac 1 {2^x} + \dfrac 1 {3^x} + \cdots$: $35.26$
* : $\S 1.2.7$: Harmonic Numbers: $(6)$
* : $\S 21$: Series of Constants: Series Involving Reciprocals of Powers of Positive Integers: $21.35.$
Category:Riemann Zeta Function at Even Integers
Riemann Zeta Function at Even Integers/Corollary

== Corollary to Riemann Zeta Function at Even Integers ==
where:
:$B_n$ are the Bernoulli numbers
:$n$ is a positive integer.
== Proof ==
== Also presented as ==
== Sources ==
* : $\S 21$: Series involving Bernoulli and Euler Numbers: $21.8$
* : $\S 23$: Bernoulli and Euler Numbers: Series involving Bernoulli and Euler Numbers: $23.8.$
Category:Riemann Zeta Function at Even Integers
Riemann Zeta Function at Even Integers/Corollary/Also presented as

== : Also presented as == 
 can also be presented in the form:
where ${B_n}^*$ denotes the archaic form of the Bernoulli numbers.
== Sources ==
* : $\S 21$: Series involving Bernoulli and Euler Numbers: $21.8$
* : $\S 23$: Bernoulli and Euler Numbers: Series involving Bernoulli and Euler Numbers: $23.8.$
Category:Riemann Zeta Function at Even Integers
Riemann Zeta Function at Even Integers/Examples

== Examples of Riemann Zeta Function at Even Integers ==
=== Riemann Zeta Function of $2$ ===
=== Riemann Zeta Function of $4$ ===
=== Riemann Zeta Function of $6$ ===
=== Riemann Zeta Function of $8$ ===
=== Riemann Zeta Function of $26$ ===
Category:Riemann Zeta Function at Even Integers
Riemann Zeta Function at Even Integers/Examples/2

== Example of Riemann Zeta Function at Even Integers ==
The Riemann zeta function of $2$ is given by:
== Proof ==
The decimal expansion can be found by an application of arithmetic.
== Sources ==
* : $1 \cdotp 644 \, 934 \, 066 \ldots$
* : $\S 1.2.7$: Harmonic Numbers: $(7)$
* : $1 \cdotp 64493 \, 4066 \ldots$
Category:Riemann Zeta Function at Even Integers
Riemann Zeta Function at Even Integers/Examples/26

== Example of Riemann Zeta Function at Even Integers ==
The Riemann zeta function of $26$ is given by:
== Proof ==
The decimal expansion can be found by an application of arithmetic.
== Sources ==
* : $3 \cdotp 14159 \, 26535 \, 89793 \, 23846 \, 26433 \, 83279 \, 50288 \, 41972 \ldots$
* : $3 \cdotp 14159 \, 26535 \, 89793 \, 23846 \, 26433 \, 83279 \, 50288 \, 41971 \ldots$
Category:Riemann Zeta Function at Even Integers
Riemann Zeta Function at Even Integers/Historical Note

== Historical Note on Riemann Zeta Function at Even Integers ==
 calculated the sums of all the even powers of the reciprocals of the integers, all the way up to the $26$th power.
== Sources ==
* : $3 \cdotp 14159 \, 26535 \, 89793 \, 23846 \, 26433 \, 83279 \, 50288 \, 41972 \ldots$
* : $\S 1.2.7$: Harmonic Numbers
* : $3 \cdotp 14159 \, 26535 \, 89793 \, 23846 \, 26433 \, 83279 \, 50288 \, 41971 \ldots$
Category:Historical Notes
Riemann Zeta Function at Even Integers/Lemma

== Lemma ==
Let $x \in \R$ be such that $\size x < 1$.
Then:
:$\ds \pi x \cot {\pi x} = 1 - 2 \sum_{n \mathop = 1}^\infty \map \zeta {2 n} x^{2 n}$
where $\zeta$ denotes the Riemann zeta function.
== Proof 1 ==
== Proof 2 ==
Category:Riemann Zeta Function at Even Integers
Riemann Zeta Function at Even Integers/Lemma/Proof 2

== Lemma ==
== Proof ==
From Laurent Series Expansion for Cotangent Function:
:$\ds \pi \cot \pi z = \frac 1 z - 2 \sum_{n \mathop = 1}^\infty \map \zeta {2 n} z^{2 n - 1}$
where:
:$z \in \C$ such that $\cmod z < 1$
:$\zeta$ is the Riemann Zeta function.
Letting $x \in \R$ replace $z$, and multiplying through by $x$:
Category:Riemann Zeta Function at Even Integers
Riemann Zeta Function at Even Integers/Proof 1

==Theorem==
==Proof==
=== Lemma ===
We also have:
Equating the coefficients of $(1)$ with the expression given in the :
:$\map \zeta {2 n} = \paren {-1}^{n + 1} \dfrac {B_{2 n} 2^{2 n - 1} \pi^{2 n} } {\paren {2 n}!}$
Riemann Zeta Function at Even Integers/Proof 2

== Theorem ==
== Proof ==
Let $k \in \N$.
Let $\map S x$ be equal to $x^{2 k}$ on $\closedint {-\pi} \pi$ and be periodic with period $2 \pi$.
Let $\ds \map I {2 m, n} = \int_0^\pi x^{2 m} \cos n x \rd x$.
Let $\map A {2 m, n} = \dfrac {\pi^{2 m - 1} \paren {-1}^n 2 m} {n^2}$.
Let $\map B {2 m, n} = -\dfrac {2 m \paren {2 m - 1} } {n^2}$.
By Fourier Series for Even Function over Symmetric Range:
We have:
Thus:
From the above: 
:$\map \zeta 2 = \dfrac {\pi^2} 6$
which serves as our base case.
Assume the induction hypothesis that, for $1 \le m \le k - 1$:
:$\map \zeta {2 m} = \paren {-1}^{m + 1} \dfrac {B_{2 m} 2^{2 m - 1} \pi^{2 m} } {\paren {2 m}!}$
Then: 
 
 
 
 
 
 
which completes the induction step.
Thus by Proof by Mathematical Induction, for all $n \ge 1$:
:$\map \zeta {2 n} = \paren {-1}^{n + 1} \dfrac {B_{2 n} 2^{2 n - 1} \pi^{2 n} } {\paren {2 n}!}$
Category:Riemann Zeta Function at Even Integers
Riemann Zeta Function at Non-Positive Integers

== Theorem == 
 
Let $n \ge 0$ be a integer. 
Then: 
:$\map \zeta {-n} = \paren {-1}^n \dfrac {B_{n + 1} } {n + 1}$ 
where:
:$B_n$ is the $n$th Bernoulli number
:$\zeta$ is the Riemann Zeta function
== Proof == 
By Hankel Representation of Riemann Zeta Function: 
:$\ds \map \zeta {-n} = \frac {i \map \Gamma {1 + n} } {2 \pi} \oint_C \frac 1 {z^{n + 1} \paren {e^z - 1} } \rd z$ 
where $C$ is the Hankel contour.
Note that the integrand is meromorphic, with exactly one pole at $z = 0$ lying inside the contour.
So: 
By the definition of a residue, the residue at $0$ is given by the coefficient of the $\dfrac 1 z$ term.
This is the term where $k = n + 1$, so:
== Examples ==
Category:Bernoulli Numbers
Category:Riemann Zeta Function
Riemann Zeta Function at Non-Positive Integers/Examples/Zeta(-1)

== Example of Use of Riemann Zeta Function at Non-Positive Integers ==
:$\map \zeta {-1 } = -\dfrac 1 {12} $
== Proof ==
Follows directly from the Riemann Zeta Function at Non-Positive Integers:
Explicit derivation illustrated below:
Category:Examples of Riemann Zeta Function
Riemann Zeta Function at Non-Positive Integers/Examples/Zeta(-3)

== Example of Use of Riemann Zeta Function at Non-Positive Integers ==
:$\map \zeta {-3 } = \dfrac 1 {120} $
== Proof ==
Follows directly from the Riemann Zeta Function at Non-Positive Integers:
Explicit derivation illustrated below:
Category:Examples of Riemann Zeta Function
Riemann Zeta Function at Odd Integers

== Unsolved Problem ==
The value of the Riemann $\zeta$ function for odd integers has not been determined.
For certain values it has been calculated to whatever number of places, but an exact expression has never been found.
== Examples ==
== Also see ==
* Riemann Zeta Function at Even Integers
== Sources ==
* : $2$
* : $2$
Category:Unsolved Problems
Category:Riemann Zeta Function
Category:Riemann Zeta Function at Odd Integers
Riemann Zeta Function in terms of Dirichlet Eta Function

== Theorem ==
Let $\zeta$ be the Riemann zeta function.
Let $\eta$ be the Dirichlet eta function.
Let $s \in \C$ be a complex number with real part $\sigma > 1$.
Then:
:$\map \zeta s = \dfrac 1 {1 - 2^{1 - s} } \map \eta s$
== Proof ==
== Also see ==
* Analytic Continuation of Riemann Zeta Function using Dirichlet Eta Function
Category:Riemann Zeta Function
Category:Dirichlet Eta Function
Riemann Zeta Function of 1000

== Theorem ==
To at least $100$ decimal places:
:$\map \zeta {1000} \approx 1$
where $\zeta$ denotes the Riemann zeta function.
== Proof ==
By definition of the general harmonic numbers:
:$\ds \map \zeta r = \lim_{n \mathop \to \infty} H_n^{\paren r} = \sum_{k \mathop \ge 1} \frac 1 {k^r}$
From Sequence of General Harmonic Numbers Converges for Index Greater than 1:
which is $1 \cdot 000 \ldots$ to a good few hundred decimal places.
== Sources ==
* : $\S 1.2.7$: Harmonic Numbers: Exercise $12$
Category:Riemann Zeta Function
Riemann Zeta Function of 4

== Theorem ==
The Riemann zeta function of $4$ is given by:
== Proof 1 ==
== Proof 2 == 
== Proof 3 ==
== Proof 4 ==
== Proof 5 ==
The decimal expansion can be found by an application of arithmetic.
== Historical Note ==
== Sources ==
* : $\S 19$: Series involving Reciprocals of Powers of Positive Integers: $19.20$
* : $1,08232 3237 \ldots$
* : $1 \cdotp 082 \, 323 \ldots$
* 
* : Chapter $\text {A}.21$: Euler ()
* : $\S 1.2.7$: Harmonic Numbers: $(7)$
* : $1 \cdotp 08232 \, 3 \ldots$
* 
* 
* : $\S 21$: Series of Constants: Series Involving Reciprocals of Powers of Positive Integers: $21.20.$
* 
Category:Riemann Zeta Function of 4
Category:Riemann Zeta Function at Even Integers‎
Category:Formulas for Pi
Category:Fourth Powers
Riemann Zeta Function of 4/Mistake

== Source Work ==
:
:Thème et variations
::$1,08232 3237 \ldots$
== Mistake ==
The section title is given as:
:$1,08232 3237 \ldots$
== Correction ==
There is a $3$ missing.
Riemann Zeta Function of 4 is $\dfrac {\pi^4} {90}$, which actually evaluates to:
:$1 \cdotp 08232 \, 323 \mathbf 3 7 \, 11138 \, 19151 \, 60036 \, 96541 \, 16790 \, 27747 \, 50951 \, 91872 \ldots$
== Sources ==
* : $1,08232 3237 \ldots$
Category:Mistakes/Les Nombres Remarquables
Riemann Zeta Function of 4/Proof 1

== Theorem ==
== Proof == 
By Fourier Series of Fourth Power of x, for $x \in \closedint {-\pi} \pi$:
:$\ds x^4 = \frac {\pi^4} 5 + \sum_{n \mathop = 1}^\infty \frac {8 n^2 \pi^2 - 48} {n^4} \map \cos {n \pi} \map \cos {n x}$
Setting $x = \pi$: 
Category:Riemann Zeta Function of 4
Riemann Zeta Function of 4/Proof 2

== Theorem ==
== Proof == 
By Fourier Series of x squared, for $x \in \closedint {-\pi} \pi$:
:$\ds x^2 = \frac {\pi^2} 3 + \sum_{n \mathop = 1}^\infty \paren {\paren {-1}^n \frac 4 {n^2} \cos n x}$
Hence:
Category:Riemann Zeta Function of 4
Riemann Zeta Function of 4/Proof 5

== Theorem ==
== Proof == 
Create a multiplication table where the column down the and the row across the top each contains the terms of zeta function of $2$:
:$\begin {array} {c|cccccccccc}
\paren {\map \zeta 2}^2 & \paren {\dfrac 1 {1^2} } & \paren {\dfrac 1 {2^2} } & \paren {\dfrac 1 {3^2} } & \paren {\dfrac 1 {4^2} } & \cdots \\
\hline
\paren {\dfrac 1 {1^2} } & \paren {\dfrac 1 {1^4} } & \paren {\dfrac 1 {1^2} } \paren {\dfrac 1 {2^2} } & \paren {\dfrac 1 {1^2} } \paren {\dfrac 1 {3^2} } & \paren {\dfrac 1 {1^2} } \paren {\dfrac 1 {4^2} } & \cdots \\
\paren {\dfrac 1 {2^2} } & \paren {\dfrac 1 {2^2} } \paren {\dfrac 1 {1^2} } & \paren {\dfrac 1 {2^4} } & \paren {\dfrac 1 {2^2} } \paren {\dfrac 1 {3^2} } & \paren {\dfrac 1 {2^2} } \paren {\dfrac 1 {4^2} } & \cdots \\
\paren {\dfrac 1 {3^2} } & \paren {\dfrac 1 {3^2} } \paren {\dfrac 1 {1^2} } & \paren {\dfrac 1 {3^2} } \paren {\dfrac 1 {2^2} } & \paren {\dfrac 1 {3^4} } & \paren {\dfrac 1 {3^2} } \paren {\dfrac 1 {4^2} } & \cdots \\
\paren {\dfrac 1 {4^2} } & \paren {\dfrac 1 {4^2} } \paren {\dfrac 1 {1^2} } & \paren {\dfrac 1 {4^2} } \paren {\dfrac 1 {2^2} } & \paren {\dfrac 1 {4^2} } \paren {\dfrac 1 {3^2} } & \paren {\dfrac 1 {4^4} } & \cdots \\
\vdots & \vdots & \vdots & \vdots & \vdots & \ddots \\
\end {array}$
The sum of all of the entries in this table is equal to $\paren {\map \zeta 2}^2$.
:$\map \zeta 4$ is the sum of the entries along the main diagonal.
We have:
Let:
Therefore:
We make the following observations:
:$(1): \quad$ The number of terms added to calculate the coefficient of the $x^3$ term is $\dbinom k 1 = k$
:$(2): \quad$ The number of terms added to calculate the coefficient of the $x^5$ term is $\dbinom k 2$
:$(3): \quad$ For $k \ge 1$, the coefficient of $x^3$ in $\ds P_k = - \dfrac 1 {\pi^2} \sum_{i \mathop = 1}^k \dfrac 1 {i^2}$
:$(4): \quad$ For $k \ge 2$, the coefficient of $x^5$ in $\ds P_k = \dfrac 1 {\pi^4} \sum_{i \mathop = 1}^{k - 1} \sum_{j \mathop = {i + 1} }^k \paren {\frac 1 {i^2} } \paren {\frac 1 {j^2} } $
Expanding the product out to k, we get:
Now recall the following two representations of the Sine of x:
Notice that by taking the limit of $P_k$ as $k \to \infty$, we obtain precisely the Euler Formula for Sine Function.
Equating the coefficient of $x^5$ in the Euler Formula for Sine Function with the Power Series Expansion for Sine Function, we have:
:$\ds \lim_{k \mathop \to \infty} \dfrac 1 {\pi^4} \sum_{i \mathop = 1}^{k - 1} \sum_{j \mathop = {i + 1} }^k \paren {\frac 1 {i^2} } \paren {\frac 1 {j^2} } = \frac 1 {5!}$
Therefore:
:$\ds \sum_{i \mathop = 1}^\infty \sum_{j \mathop = {i + 1} }^\infty \paren {\frac 1 {i^2} } \paren {\frac 1 {j^2} } = \frac {\pi^4} {5!}$
Therefore:
Category:Riemann Zeta Function of 4
Riemann Zeta Function of 6

== Theorem ==
The Riemann zeta function of $6$ is given by:
== Proof 1 ==
== Proof 2 ==
== Proof 3 ==
== Historical Note ==
== Sources ==
* : $\S 19$: Series involving Reciprocals of Powers of Positive Integers: $19.21$
* 
* : Chapter $\text {A}.21$: Euler ()
* : $\S 1.2.7$: Harmonic Numbers: $(7)$
* : $\S 21$: Series of Constants: Series Involving Reciprocals of Powers of Positive Integers: $21.21.$
* 
Category:Riemann Zeta Function of 6
Category:Riemann Zeta Function at Even Integers
Category:Formulas for Pi
Category:Sixth Powers
Riemann Zeta Function of 6/Historical Note

== Historical Note on Riemann Zeta Function of 6 ==
The Riemann Zeta Function of 6 was solved by , using the same technique as for the Riemann Zeta Function of 4 and the Basel Problem.
:If only my brother were alive now.
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== Sources ==
* : Chapter $\text {A}.21$: Euler ()
Category:Historical Notes