Let Sn=1n1+2n1+21+3n1+21+31+⋯.
Then, for positive even numbers m, there is a beautiful relationship between Sm and the Riemann zeta function ζ(⋅):
S2S4S6S8Sm====⋮=2ζ(3)3ζ(5)−ζ(2)ζ(3)4ζ(7)−ζ(2)ζ(5)−ζ(3)ζ(4)5ζ(9)−ζ(2)ζ(7)−ζ(3)ζ(6)−ζ(4)ζ(5)2m+2ζ(m+1)−k=2∑2mζ(k)ζ(m−k).
However, there is also a relationship between positive odd numbers m and the Riemann zeta function. Find this relationship and submit your answer as S3π4.
Bonus: Prove the pattern shown above.