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 $x \in \closedint {-\pi} \pi$:
- $\displaystyle x^4 = \frac {\pi^4} 5 + \sum_{n \mathop = 1}^\infty \frac {8 n^2 \pi^2 - 48} {n^4} \, \map \cos {n \pi} \, \map \cos {n x}$
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 $x \in \left[{- \pi \,.\,.\, \pi}\right]$:
- $\displaystyle x^2 = \frac {\pi^2} 3 + \sum_{n \mathop = 1}^\infty \left({\left({-1}\right)^n \frac 4 {n^2} \cos n x}\right)$
Hence:
\(\ds \frac 1 \pi \int_{-\pi}^\pi x^4 \, \mathrm d x\) | \(=\) | \(\ds \frac 1 2 \left({\frac {2 \pi^2} 3}\right)^2 + \sum_{n \mathop = 1}^\infty \left({\frac {4 \left({-1}\right)^n} {n^2} }\right)^2\) | Parseval's Theorem | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac 2 \pi \int_0^\pi x^4 \, \mathrm d x\) | \(=\) | \(\ds \frac {2 \pi^4} 9 + \sum_{n \mathop = 1}^\infty \frac {16} {n^4}\) | Definite Integral of Even Function | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {2 \pi^4} 5\) | \(=\) | \(\ds \frac {2 \pi^4} 9 + \sum_{n \mathop = 1}^\infty \frac {16} {n^4}\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {8 \pi^4} {45}\) | \(=\) | \(\ds \sum_{n \mathop = 1}^\infty \frac {16} {n^4}\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \sum_{n \mathop = 1}^\infty \frac 1 {n^4}\) | \(=\) | \(\ds \frac {\pi^4} {90}\) |
$\blacksquare$
Proof 3
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 of two
- $\begin{array}{r|cccccccccc} \displaystyle \paren {\map \zeta 2}^2 & \paren {\dfrac {1} {1^2}} & \paren {\dfrac {1} {2^2}} & \paren {\dfrac {1} {3^2}} & \paren {\dfrac {1} {4^2}} & \cdots \\ \hline \paren {\dfrac {1} {1^2}} & \paren {\dfrac {1} {1^4}} & \paren {\dfrac {1} {1^2}} \paren {\dfrac {1} {2^2}} & \paren {\dfrac {1} {1^2}} \paren {\dfrac {1} {3^2}} & \paren {\dfrac {1} {1^2}} \paren {\dfrac {1} {4^2}} & \cdots \\ \paren {\dfrac {1} {2^2}} & \paren {\dfrac {1} {2^2}} \paren {\dfrac {1} {1^2}} & \paren {\dfrac {1} {2^4}} & \paren {\dfrac {1} {2^2}} \paren {\dfrac {1} {3^2}} & \paren {\dfrac {1} {2^2}} \paren {\dfrac {1} {4^2}} & \cdots \\ \paren {\dfrac {1} {3^2}} & \paren {\dfrac {1} {3^2}} \paren {\dfrac {1} {1^2}} & \paren {\dfrac {1} {3^2}} \paren {\dfrac {1} {2^2}} & \paren {\dfrac {1} {3^4}} & \paren {\dfrac {1} {3^2}} \paren {\dfrac {1} {4^2}} & \cdots \\ \paren {\dfrac {1} {4^2}} & \paren {\dfrac {1} {4^2}} \paren {\dfrac {1} {1^2}} & \paren {\dfrac {1} {4^2}} \paren {\dfrac {1} {2^2}} & \paren {\dfrac {1} {4^2}} \paren {\dfrac {1} {3^2}} & \paren {\dfrac {1} {4^4}} & \cdots \\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ \end{array}$
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 | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{i \mathop = 1}^{\infty} {\frac 1 {i^4 } } + \sum_{i \mathop = 2}^{\infty} \sum_{j \mathop = 1}^{i-1} {\frac 1 {i^2 } } {\frac 1 {j^2 } } + \sum_{j \mathop = 2}^{\infty} \sum_{i \mathop = 1}^{j - 1} {\frac 1 {i^2 } } {\frac 1 {j^2 } }\) | $\paren {i = j } + \paren {j \lt i } + \paren {j \gt i }$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \zeta 4 + 2 \dfrac {\pi^4} {5!}\) | The sums $\paren {j < i }$ and $\paren {j > i }$ are symmetric and each equal to the coefficient of the 5th power term in the sin(x) expansion See Basel_Problem/Proof_2 |
Therefore:
\(\ds \map \zeta 4\) | \(=\) | \(\ds \paren {\map \zeta 2}^2 - 2 \dfrac {\pi^4} {5!}\) | Rearranging | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\pi^4} {36} - \dfrac {\pi^4} {60}\) | Basel Problem | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\pi^4} {90}\) | simplifying |
$\blacksquare$
The decimal expansion can be found by an application of arithmetic.
Historical Note
The Riemann Zeta Function of 4 was solved by Leonhard Euler, using the same technique as for the Basel Problem.
- If only my brother were alive now.
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): $1 \cdotp 082 \, 323 \ldots$
- 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): $1 \cdotp 08232 \, 3 \ldots$