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:

\(\ds \map \zeta {-n}\)

\(=\)

\(\ds \paren {-1}^n \frac {B_{n + 1} } {n + 1}\)

Riemann Zeta Function at Non-Positive Integers

\(\ds \)

\(=\)

\(\ds \paren {-1}^3 \frac {B_{3 + 1} } {3 + 1}\)

Entering $n = 3$

\(\ds \)

\(=\)

\(\ds -\frac {B_{4} } {4}\)

\(\ds \)

\(=\)

\(\ds \frac 1 {120}\)

From Definition:Bernoulli Numbers/Sequence, $B_4 = -\dfrac 1 {30} $

$\blacksquare$