Riemann Zeta Function of 4

Theorem

The Riemann zeta function of $4$ is given by:

$\ds \map \zeta 4$

$=$

$\ds \dfrac 1 {1^4} + \dfrac 1 {2^4} + \dfrac 1 {3^4} + \dfrac 1 {4^4} + \cdots$

$\ds $

$=$

$\ds \dfrac {\pi^4} {90}$

$\ds $

$\approx$

$\ds 1 \cdotp 08232 \, 3 \ldots$

This sequence is A013662 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

Proof 1

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 $x = \pi$:

$\ds \pi^4$

$=$

$\ds \frac {\pi^4} 5 + \sum_{n \mathop = 1}^\infty \frac {8 n^2 \pi^2 - 48} {n^4} \map {\cos^2} {n \pi}$

$\ds \leadsto \ \ $

$\ds \frac {4 \pi^4} 5$

$=$

$\ds \sum_{n \mathop = 1}^\infty \frac {8 n^2 \pi^2} {n^4} - \sum_{n \mathop = 1}^\infty \frac {48} {n^4}$

Cosine of Multiple of Pi

$\ds \leadsto \ \ $

$\ds \frac {\pi^4} 5$

$=$

$\ds 2 \pi^2 \sum_{n \mathop = 1}^\infty \frac 1 {n^2} - 12 \sum_{n \mathop = 1}^\infty \frac 1 {n^4}$

$\ds $

$=$

$\ds \frac {\pi^4} 3 - 12 \sum_{n \mathop = 1}^\infty \frac 1 {n^4}$

Basel Problem

$\ds \leadsto \ \ $

$\ds 12 \sum_{n \mathop = 1}^\infty \frac 1 {n^4}$

$=$

$\ds \frac {\pi^4} 3 - \frac {\pi^4} 5$

rearranging

$\ds $

$=$

$\ds \frac {2 \pi^4} {15}$

$\ds \leadsto \ \ $

$\ds \sum_{n \mathop = 1}^\infty \frac 1 {n^4}$

$=$

$\ds \frac {\pi^4} {90}$

$\blacksquare$

Proof 2

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:

$\ds \frac 1 \pi \int_{-\pi}^\pi x^4 \rd x$

$=$

$\ds \frac 1 2 \paren {\frac {2 \pi^2} 3}^2 + \sum_{n \mathop = 1}^\infty \paren {\frac {4 \paren {-1}^n} {n^2} }^2$

Parseval's Theorem

$\ds \leadsto \ \ $

$\ds \frac 2 \pi \int_0^\pi x^4 \rd x$

$=$

$\ds \frac {2 \pi^4} 9 + \sum_{n \mathop = 1}^\infty \frac {16} {n^4}$

Definite Integral of Even Function

$\ds \leadsto \ \ $

$\ds \frac {2 \pi^4} 5$

$=$

$\ds \frac {2 \pi^4} 9 + \sum_{n \mathop = 1}^\infty \frac {16} {n^4}$

$\ds \leadsto \ \ $

$\ds \frac {8 \pi^4} {45}$

$=$

$\ds \sum_{n \mathop = 1}^\infty \frac {16} {n^4}$

$\ds \leadsto \ \ $

$\ds \sum_{n \mathop = 1}^\infty \frac 1 {n^4}$

$=$

$\ds \frac {\pi^4} {90}$

$\blacksquare$

This theorem requires a proof.
In particular: $\ds \int_0^1\int_0^1\int_0^1\int_0^1 \frac 1 {1 - x_1x_2x_3x_4} \rd x_1 \rd x_2 \rd x_3 \rd x_4$. Use Beukers-Calabi-Kolk sub.
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{ProofWanted}} from the code.
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Proof 4

$\ds \map \zeta 4$

$=$

$\ds \paren {-1}^3 \dfrac {B_4 2^3 \pi^4} {4!}$

Riemann Zeta Function at Even Integers

$\ds $

$=$

$\ds \paren {-1}^3 \paren {-\dfrac 1 {30} } \dfrac {2^3 \pi^4} {4!}$

Definition of Sequence of Bernoulli Numbers

$\ds $

$=$

$\ds \paren {\dfrac 1 {30} } \paren {\dfrac 8 {24} } \pi^4$

Definition of Factorial

$\ds $

$=$

$\ds \dfrac {\pi^4} {90}$

simplifying

$\blacksquare$

Proof 5

Create a multiplication table where the column down the left hand side 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:

$\ds \paren {\map \zeta 2}^2$

$=$

$\ds \paren {\sum_{i \mathop = 1}^\infty {\frac 1 {i^2} } } \paren {\sum_{j \mathop = 1}^\infty {\frac 1 {j^2} } }$

$\ds $

$=$

$\ds \sum_{i \mathop = 1}^\infty \sum_{j \mathop = 1}^\infty {\frac 1 {i^2} } {\frac 1 {j^2} }$

Product of Absolutely Convergent Series

$\ds $

$=$

$\ds \sum_{i \mathop = 1}^{\infty} {\frac 1 {i^4} } + \sum_{i \mathop = 1}^\infty \sum_{j \mathop = {i + 1} }^\infty {\frac 1 {i^2} } {\frac 1 {j^2} } + \sum_{j \mathop = 1}^\infty \sum_{i \mathop = {j + 1} }^\infty {\frac 1 {i^2} } {\frac 1 {j^2} }$

$\paren {i = j} + \paren {j > i} + \paren {j < i}$

$\ds $

$=$

$\ds \map \zeta 4 + 2 \sum_{i \mathop = 1}^\infty \sum_{j \mathop = {i + 1} }^\infty {\frac 1 {i^2} } {\frac 1 {j^2} }$

Let:

$\ds P_k$

$=$

$\ds x \prod_{n \mathop = 1}^k \paren {1 - \frac {x^2} {n^2 \pi^2} }$

$\ds $

$=$

$\ds x \paren {1 - \dfrac {x^2} {1 \pi^2} } \paren {1 - \dfrac {x^2} {2^2 \pi^2} } \paren {1 - \dfrac {x^2} {3^2 \pi^2} } \cdots \paren {1 - \dfrac {x^2} {k^2 \pi^2} }$

Therefore:

$\ds P_1$

$=$

$\ds x - \frac {x^3} {\pi^2} \paren {\dfrac 1 {1^2} }$

$\ds P_2$

$=$

$\ds x - \frac {x^3} {\pi^2} \paren {\dfrac 1 {1^2} + \dfrac 1 {2^2} } + \frac {x^5} {\pi^4} \paren {\dfrac 1 {1^2} \dfrac 1 {2^2} }$

$\ds P_3$

$=$

$\ds x - \frac {x^3} {\pi^2} \paren {\dfrac 1 {1^2} + \dfrac 1 {2^2} + \dfrac 1 {3^2} } + \frac {x^5} {\pi^4} \paren {\dfrac 1 {1^2} \dfrac 1 {2^2} + \dfrac 1 {1^2} \dfrac 1 {3^2} + \dfrac 1 {2^2} \dfrac 1 {3^2} } - \frac {x^7} {\pi^6} \paren {\dfrac 1 {1^2} \dfrac 1 {2^2} \dfrac 1 {3^2} }$

$\ds P_4$

$=$

$\ds x - \frac {x^3} {\pi^2} \paren {\dfrac 1 {1^2} + \dfrac 1 {2^2} + \dfrac 1 {3^2} + \dfrac 1 {4^2} } + \frac {x^5} {\pi^4} \paren {\dfrac 1 {1^2} \dfrac 1 {2^2} + \dfrac 1 {1^2} \dfrac 1 {3^2} + \dfrac 1 {1^2} \dfrac 1 {4^2} + \dfrac 1 {2^2} \dfrac 1 {3^2} + \dfrac 1 {2^2} \dfrac 1 {4^2} + \dfrac 1 {3^2} \dfrac 1 {4^2} } - \frac {x^7} {\pi^6} \paren {\dfrac 1 {1^2} \dfrac 1 {2^2} \dfrac 1 {3^2} + \dfrac 1 {1^2} \dfrac 1 {2^2} \dfrac 1 {4^2} + \dfrac 1 {1^2} \dfrac 1 {3^2} \dfrac 1 {4^2} + \dfrac 1 {2^2} \dfrac 1 {3^2} \dfrac 1 {4^2} } + \frac {x^9} {\pi^8} \paren {\dfrac 1 {1^2} \dfrac 1 {2^2} \dfrac 1 {3^2} \dfrac 1 {4^2} }$

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:

$\ds P_k$

$=$

$\ds x - \dfrac {x^3} {\pi^2} \sum_{i \mathop = 1}^k \dfrac 1 {i^2} + \frac {x^5} {\pi^4} \sum_{i \mathop = 1}^{k - 1} \sum_{j \mathop = {i + 1} }^k \paren {\frac 1 {i^2} } \paren {\frac 1 {j^2} } - \cdots$

Now recall the following two representations of the Sine of x:

$\ds \sin x$

$=$

$\ds x \prod_{n \mathop = 1}^\infty \paren {1 - \dfrac {x^2} {n^2 \pi^2} }$

Euler Formula for Sine Function

$\ds \sin x$

$=$

$\ds \sum_{n \mathop = 0}^\infty \paren {-1}^n \dfrac {x^{2 n + 1} } {\paren {2 n + 1}!} = x - \dfrac {x^3} {3!} + \dfrac {x^5} {5!} - \dfrac {x^7} {7!} + \cdots$

Power Series Expansion for Sine Function

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:

$\ds \paren {\map \zeta 2}^2$

$=$

$\ds \map \zeta 4 + 2 \sum_{i \mathop = 1}^\infty \sum_{j \mathop = {i + 1} }^\infty {\frac 1 {i^2} } {\frac 1 {j^2} }$

$\ds \paren {\map \zeta 2}^2$

$=$

$\ds \map \zeta 4 + 2 \dfrac {\pi^4} {5!}$

$\ds \map \zeta 4$

$=$

$\ds \paren {\map \zeta 2}^2 - 2 \dfrac {\pi^4} {5!}$

Rearranging

$\ds $

$=$

$\ds \paren{\dfrac {\pi^2} 6 }^2 - \dfrac {\pi^4} {60}$

Basel Problem

$\ds $

$=$

$\ds \dfrac {\pi^4} {36} - \dfrac {\pi^4} {60}$

$\ds $

$=$

$\ds \dfrac {5\pi^4} {180} - \dfrac {3\pi^4} {180}$

$\ds $

$=$

$\ds \frac {2\pi^4} {180}$

$\ds $

$=$

$\ds \frac {\pi^4} {90}$

$\blacksquare$

The decimal expansion can be found by an application of arithmetic.

Historical Note

The was solved by Leonhard Euler, using the same technique as for the Basel Problem.

If only my brother were alive now.

-- Johann Bernoulli

Sources

1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 19$: Series involving Reciprocals of Powers of Positive Integers: $19.20$
1983: François Le Lionnais and Jean Brette: Les Nombres Remarquables ... (previous) ... (next): $1,08232 3237 \ldots$
1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $1 \cdotp 082 \, 323 \ldots$
1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): zeta function
1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {A}.21$: Euler ($\text {1707}$ – $\text {1783}$)
1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.7$: Harmonic Numbers: $(7)$
1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $1 \cdotp 08232 \, 3 \ldots$
1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Riemann zeta function
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Riemann zeta function
2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 21$: Series of Constants: Series Involving Reciprocals of Powers of Positive Integers: $21.20.$
2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): zeta function

Theorem

Proof 1

Proof 2

Proof 3

Proof 4

Proof 5

Historical Note

Sources