Is this a new representation of (some) Bernoulli numbers?

Question

Let $\operatorname{B}(n)$ denote the Bernoulli numbers and $\operatorname{b}(n) = \operatorname{B}(n)/n$ with $b(0)=1$ the divided Bernoulli numbers. Also let $\sigma_{k}(n)= \sum_{d \mid n} d^k$ denote the sum of the $k$-th powers of the positive divisors of $n$.

Recently Simon Plouffe made the following conjecture in OEIS A013965:

$$ \sum_{j \ge 1} \frac{\sigma_{17}(j)}{\exp(2 \pi j)} \, = \, \frac{43867}{28728} \, = \, \frac{\operatorname{B}(18)}{36} . $$

Plouffe found similar relations earlier, but without mentioning the connection with the Bernoulli numbers, for example in A001160 and A013961.

Vaclav Kotesovec generalized these observations in A362870 with the following far-reaching conjecture:

If $\nu$ is twice an odd number greater than 1 (i.e., $\nu = 4n+2$, a term of A016825 that is greater than 2), then

$$ \operatorname{b}(\nu) = \frac{\operatorname{B}(\nu)}{\nu} \, = \, 2 \sum_{j \ge 1} \frac{\sigma_{\nu-1}(j)} { \exp(2 \pi j)} \, = \, \frac{A358625(\nu)}{A075180(\nu - 1)}. $$

We numerically checked the first 50 terms $\nu > 2$ of A016825. They suggest the plausibility of the conjecture. Although it becomes increasingly challenging for larger values to achieve satisfactory accuracy, we gained in this range at least agreement in the first 800 decimal places.

Does anyone know a proof or can give references to relevant literature?

Edit: From Raymond Manzoni's answer and the literature he cites it is clear that the formulas quoted above are not conjectures but proven facts. A detailed presentation of the theory and a proof of Ramanujan's theorem can be found in:

Bruce C. Berndt, Analytic Eisenstein series, theta functions, and series relations in the spirit of Ramanujan. J. Reine Angew. Math., 303/304:332–365, 1978. (See Corollary 5.10, p.355).

Answer 1

There are a lot of connections with those sums and Bernoulli numbers. That result I published on the OEIS about A013965, A001160 and A013961 is not isolated. You can see others like that here: https://vixra.org/pdf/2305.0022v2.pdf. There are 659 other results in the same vein. Most of those are not yet published on the OEIS.

Now about the function sigma and related sums: there are other results also but this time connected to prime numbers and the Eisenstein series here: https://vixra.org/pdf/2304.0180v2.pdf.

These were inspired by a finding from Ramanujan about Bernoulli numbers and this sum :

$$ \sum_{n>=1}\ \frac{n^{4 k+1}}{\exp(2 \pi n) - 1} = \frac{\operatorname{Bernoulli}(4 k+2)}{2(4k+2)}$$ I extended the finding to other sums. Does this put some light on these sums ? Best regards, Simon Plouffe

Answer 2

Simon Plouffe proposed in this $2003$ article (and in his answer here) this formula implicit in Ramanujan's work : $$\frac {B_{4n+2}}{8n+4}=\sum_{k=1}^\infty \frac {k^{4n+1}}{e^{\,2k\pi}-1}$$

According to Bruce Berndt in "Ramanujan's Notebooks part V" p.$427$ (see too his $1977$ paper "Modular transformations and generalizations of several formulae of Ramanujan" ) this formula was proved first by Glaisher in $1889$ although established by Hurwitz in $1881$ in his thesis.
Other interesting generalizations may be found in Vepsas "On Plouffe's Ramanujan Identities" and Bagis "Evaluations of Series Related to Jacobi Elliptic Functions" as well as Zucker's "The Summation of Series of Hyperbolic Functions" and others (for people hoping to generalize this for other values of $\zeta$...).

Following Lambert series is a generating function for the $\sigma_a$ function : $$\sum_{k=1}^\infty \frac{k^a q^k}{1-q^k} = \sum_{k=1}^\infty \sum_{j=1}^\infty k^a q^{j\,k} = \sum_{k=1}^\infty q^k \sigma_a(k)$$

Use $\,a=4n+1$ and $\,q=e^{-2\pi}\,$ to conclude.