3 Growth of divided Bernoulli numbers
For every \(k \ge 1\),
\[ \left|\frac{B_{2k}}{2k}\right| \ \ge \ \frac{(2k-1)!}{2^{2k-1}\, \pi ^{2k}}. \]
Proof
Euler’s formula gives
\[ \zeta (2k) \; =\; (-1)^{k+1}\, \frac{B_{2k}\, (2\pi )^{2k}}{2\, (2k)!}, \]
and \(\zeta (2k) \ge 1\), so \(|B_{2k}| \ge 2\, (2k)!/(2\pi )^{2k}\). Dividing by \(2k\) yields the claim.
\(\left|B_{2k}/2k\right| \to \infty \) as \(k \to \infty \).
Proof
The lower bound of theorem 3.1 tends to infinity: factorial growth beats the geometric factor \((2\pi ^2)^{k}\) in the denominator.
For every positive even \(C\) there is a \(t\) with \(\left|B_{m}/m\right| {\gt} 1\) for \(m = C\cdot 2^t\).
Proof
The indices \(C \cdot 2^t\) are even and tend to infinity, so theorem 3.2 applies along this subsequence.