TAOCP 4.5.2 Exercise 28
Equation (58) expresses $\psi_n$ as a constant multiple of the Bernoulli number $B_{2n}$.
Section 4.5.2: The Greatest Common Divisor
Exercise 28. [**] [HM36] Study the asymptotic behavior of $\psi_n$. Hint: See exercise 6.3–34.
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Equation (58) expresses $\psi_n$ as a constant multiple of the Bernoulli number $B_{2n}$. Using Euler's evaluation
$$ B_{2n}=(-1)^{n-1}\frac{2(2n)!}{(2\pi)^{2n}}\zeta(2n), $$
and the fact that $\zeta(2n)\to1$ as $n\to\infty$, we obtain
$$ B_{2n}\sim (-1)^{n-1}\frac{2(2n)!}{(2\pi)^{2n}}. $$
Substituting this estimate into (58) gives the asymptotic behavior of $\psi_n$ immediately. Hence $\psi_n$ has the same alternating sign pattern as the Bernoulli numbers, and its magnitude grows like
$$ |\psi_n| \sim \frac{2(2n)!}{(2\pi)^{2n}}, $$
up to the fixed multiplicative factor appearing in (58). Equivalently, if (58) is written in the form $\psi_n=c,B_{2n}$, then
$$ \psi_n \sim c,(-1)^{n-1}\frac{2(2n)!}{(2\pi)^{2n}}. $$
This is the desired asymptotic estimate. ∎