Can the formula for the sum of the first n squares be deduced from the known value of zeta of 2? I know this is asking that we swat a fly with a sledge hammer, but this off-the-wall question is prompted by the fact that they both have 6 in the denominator. That is, is that common 6 in the denominator the tip of the ice berg of a direct connection between them?

> Is that common $6$ in the denominator the tip of the ice berg of a direct connection between them?

Yes. See Bernoulli numbers. In particular, the values of $\zeta(2n)$, as well the Faulhaber's formulas for

the sums of _k_ -th powers of the first _n_ natural numbers, depend on them. In this case, $B_2=\dfrac16$