Hello! Bonjour ! ¡Hola! Guten Tag! Привет! سَّلَام!
I'm mathematician, and also a little bit YouTuber. ζ ( s ) = { ∑ n = 1 ∞ 1 n s [ ℜ e ( s ) ≥ 1 ] 2 s π s − 1 sin ( π s 2 ) Γ ( 1 − x ) ζ ( 1 − x ) [ ℜ e ( s ) < 1 ] {\displaystyle \zeta (s)={\begin{cases}\sum _{n\mathop {=} 1}^{\infty }{\frac {1}{n^{s}}}&[\Re {\mathfrak {e}}(s)\geq 1]\\2^{s}\pi ^{s\mathop {-} 1}\sin \left({\frac {\pi s}{2}}\right)\Gamma (1-x)\zeta (1-x)&[\Re {\mathfrak {e}}(s)<1]\end{cases}}}
ζ ( − 1 ) = 2 − 1 π − 2 sin ( − π 2 ) Γ ( 2 ) ζ ( 2 ) = 1 2 ( 1 π 2 ) ( − 1 ) ( 1 ) ( π 2 6 ) = − 1 2 π 2 ( π 2 6 ) = − π 2 12 π 2 = − 1 12 {\displaystyle {\begin{aligned}\zeta (-1)&=2^{-1}\pi ^{-2}\sin \left({\frac {-\pi }{2}}\right)\Gamma (2)\zeta (2)\\&={\frac {1}{2}}\left({\frac {1}{\pi ^{2}}}\right)(-1)(1)\left({\frac {\pi ^{2}}{6}}\right)\\&=-{\frac {1}{2\pi ^{2}}}\left({\frac {\pi ^{2}}{6}}\right)\\&=-{\frac {\pi ^{2}}{12\pi ^{2}}}=-{\frac {1}{12}}\end{aligned}}} e i π = cos ( π ) + i sin ( π ) = ( − 1 ) + 0 i = − 1 {\displaystyle {\begin{aligned}\mathrm {e} ^{\mathrm {i} \pi }&=\cos(\pi )+\mathrm {i} \sin(\pi )\\&=(-1)+0\mathrm {i} =-1\end{aligned}}}
q 0 = 5 + 9 i + 2 j + 6 k q 1 = 1 + 5 i + 9 j + 3 k q 0 ( q 1 ) = ( 5 + 9 i + 2 j + 6 k ) ( 1 + 5 i + 9 j + 3 k ) = ( 5 + 25 i + 45 j + 15 k ) + ( 9 i − 45 + 81 k − 27 j ) = + ( 2 j − 10 k − 18 + 6 i ) + ( 6 k + 30 j − 54 i − 18 ) = − 76 − 14 i + 50 j + 92 k {\displaystyle {\begin{aligned}q_{0}&=5+9\mathrm {i} +2\mathrm {j} +6\mathrm {k} \\q_{1}&=1+5\mathrm {i} +9\mathrm {j} +3\mathrm {k} \\\\q_{0}(q_{1})&=(5+9\mathrm {i} +2\mathrm {j} +6\mathrm {k} )(1+5\mathrm {i} +9\mathrm {j} +3\mathrm {k} )\\&=(5+25\mathrm {i} +45\mathrm {j} +15\mathrm {k} )+(9\mathrm {i} -45+81\mathrm {k} -27\mathrm {j} )\\&{\phantom {=}}+(2\mathrm {j} -10\mathrm {k} -18+6\mathrm {i} )+(6\mathrm {k} +30\mathrm {j} -54\mathrm {i} -18)\\&=-76-14\mathrm {i} +50\mathrm {j} +92\mathrm {k} \end{aligned}}}
If you're wondering, yes I listen classic music: An der schönen blauen Donau Op.314 by Johann Strauss.
![{\displaystyle \zeta (s)={\begin{cases}\sum _{n\mathop {=} 1}^{\infty }{\frac {1}{n^{s}}}&[\Re {\mathfrak {e}}(s)\geq 1]\\2^{s}\pi ^{s\mathop {-} 1}\sin \left({\frac {\pi s}{2}}\right)\Gamma (1-x)\zeta (1-x)&[\Re {\mathfrak {e}}(s)<1]\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8baacc6984d3872346847e62085693917a0a805f)
