Is the sequence of the even Bernoulli numbers bounded?

Is the sequence of the even Bernoulli numbers $B_{2n}$ bounded? And if so is it posible to define a Dirichlet series $$f(s)= \sum_{n=1}^\infty \frac{a_n}{n^s}$$ with $a_n=B_{2n}$ that is convergent at $s=1$ or that one can assign a principal value in this point?

$$B_{2n} \sim (-1)^{n-1} 4 \sqrt{\pi n} \left( \frac{n}{\pi e} \right)^{2n}$$ as can be seen from the formula for $\zeta (2n)$ and applying Sterling's formula, so the sequence of even Bernoulli numbers is unbounded.

I haven't seen a Dirichlet series for the Bernoulli numbers, but they do have an exponential generating function and a normal generating function if that helps. Another approach might be to take a look at Tao's blog post on smoothing sums to evaluate asymptotics of partial sums of divergent series. He uses it to find $-1/12$ as a constant in the expansion of $\sum_{n \leq N} n$ as well as to find values for the zeta function at other negative integers. That said, the Bernoulli numbers might grow too quickly to behave nicely (for instance, this technique fails for $\sum_{n \leq N} n!$).

EDIT: The digamma function has an asymptotic expansion of

$$\psi(x) = \ln x - \frac{1}{2x} + \sum_{n=1}^\infty \frac{B_{2n}}{2n x^{2n}}$$

that doesn't converge for any $x$, but is useful if you truncate the series to a finite number of terms. Plugging in $x=1$ to both sides yields

$$ \frac{1}{2} -\gamma = \sum_{n=1}^\infty \frac{B_{2n}}{2n}$$

though I'm not sure this makes any sense to consider.

I think that the following two interesting double inequalities \eqref{Bernoulli-ineq} and \eqref{ineq-Bernou-equiv} are useful for better answering this question.

• The double inequality \begin{equation}\label{Bernoulli-ineq}\tag{1} \frac{2(2n)!}{(2\pi)^{2n}} \frac{1}{1-2^{\alpha -2n}} \le |B_{2n}| \le \frac{2(2n)!}{(2\pi)^{2n}}\frac{1}{1-2^{\beta -2n}} \end{equation} is valid for $n\in\mathbb{N}$, where $\alpha=0$ and $ \beta=2+\frac{\ln(1-6/\pi^2)}{\ln2}=0.6491\dotsc $ are the best possible in the sense that they cannot be replaced respectively by any bigger and smaller constants. See the paper [1] below.
• The ratios $\frac{|B_{2(n+1)}|}{|B_{2n}|}$ for $n\in\mathbb{N}$ can be bounded by \begin{equation}\label{ineq-Bernou-equiv}\tag{2} \frac{2^{2n-1}-1}{2^{2n+1}-1}\frac{(2n+1)(2n+2)}{\pi^2} <\frac{|B_{2(n+1)}|}{|B_{2n}|} <\frac{2^{2n}-1}{2^{2n+2}-1}\frac{(2n+1)(2n+2)}{\pi^2}. \end{equation} See the paper [2] below.

References

1. H. Alzer, Sharp bounds for the Bernoulli numbers, Arch. Math. (Basel) 74 (2000), no. 3, 207--211; available online at https://doi.org/10.1007/s000130050432.
2. Feng Qi, A double inequality for the ratio of two non-zero neighbouring Bernoulli numbers, Journal of Computational and Applied Mathematics 351 (2019), 1--5; available online at https://doi.org/10.1016/j.cam.2018.10.049.
3. Ye Shuang, Bai-Ni Guo, and Feng Qi, Logarithmic convexity and increasing property of the Bernoulli numbers and their ratios, Revista de la Real Academia de Ciencias Exactas Fisicas y Naturales Serie A Matematicas 115 (2021), no. 3, Paper No. 135, 12 pages; available online at https://doi.org/10.1007/s13398-021-01071-x.
4. Feng Qi, Notes on a double inequality for ratios of any two neighbouring non-zero Bernoulli numbers, Turkish Journal of Analysis and Number Theory 6 (2018), no. 5, 129--131; available online at https://doi.org/10.12691/tjant-6-5-1.
5. Z.-H. Yang and J.-F. Tian, Sharp bounds for the ratio of two zeta functions, J. Comput. Appl. Math. 364 (2020), 112359, 14 pages; available online at https://doi.org/10.1016/j.cam.2019.112359.
6. L. Zhu, New bounds for the ratio of two adjacent even-indexed Bernoulli numbers, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM 114 (2020), no. 2, Paper No. 83, 13 pages; available online at https://doi.org/10.1007/s13398-020-00814-6.