Numerator in the zeta values at negative odd integers The real J-homomorphism produces cyclic subgroups of size the denominator of $\zeta(1-2k)=-B_{2k}/k$ for $k>0$ in $\pi_{4k-1}S$ which completely account for first layer of the chromatic filtration on $\pi_*S$.
Question: Does the numerator in $\zeta(1-2k)$ also relate to $\pi_*S$ (at higher layers in the chromatic filtration)?
Guess: Can we think of denominator of $\zeta(1-2k)$ as local information (since it is hit by the J-homomorphism) and the numerator of $\zeta(1-2k)$ as global information?
 A: (Too long for a comment:)
As A.S. commented, the absolute value of $\zeta(1-2k) = - B_{2k}/2k$ for $k\ge1$ is realized as twice the order of $K_{4k-2}(Z)$ divided by the order of $K_{4k-1}(Z)$, so for Bernoulli numbers the connection to the algebraic K-theory spectrum $K(Z)$ is much stronger than the connection to the sphere spectrum $S$
Implicitly completing at an odd prime $p$, $K_{4k-2}(Z)$ comes from $H_{et}^2(Z[1/p]; Z_p(2k))$ and $K_{4k-1}(Z)$ comes from $H_{et}^1(Z[1/p]; Z_p(2k))$. After (chromatic) $K(1)$-localization, $L_{K(1)} K(Z)$ is more-or-less the $G$-homotopy fixed points for an action on complex topological K-theory, $KU$, where $G$ has cohomological dimension two.
Similarly, the image-of-J spectrum $L_{K(1)} S$ is the $\bar G$-homotopy fixed points of $KU$, where $\bar G$ is the quotient of $G$ generated by Adams operations, whose cohomology realizes $H^1$ of $G$ but not $H^2$. The projection $G \to \bar G$ is compatible with the unit map $S \to K(Z)$. Now $K(Z)$ is essentially $K(1)$-local, so this suggests that both the numerator and the denominator of $\zeta(1-2k)$ is accounted for by a $K(1)$-local object. This also fits with the Kummer congruences, telling you that whether a prime $p$ divides $B_{2k}/2k$ depends on $2k$ mod $p-1$, so that any $p$-torsion in $K_{4k-2}(Z)$ will reappear (in some form) every $2p-2$ degrees.
This is the usual degree of $v_1$-periodicity.
See Dwyer-Mitchell "On the K-theory spectrum of a ring of algebraic integers" K-Theory 14 (1998), no. 3, 201–263, for more general rings of integers than $Z$.
Higher chromatic periodicities tend to repeat every $|v_n| = 2p^n-2$ degrees, which for $n\ge2$ does not match with the Kummer congruences.
However, some $v_2$-periodic and $v_3$-periodic families in $\pi_*(S)$ are known to be detected in $\pi_* K(BP\langle 1\rangle)$ and $\pi_* K(BP\langle 2\rangle)$, respectively, where the truncated Brown-Peterson spectrum $BP\langle 1\rangle$ is closely related to topological K-theory, and $BP\langle 2\rangle$ is related to elliptic cohomology and topological modular forms. See Angelini-Knoll-Ausoni-Culver-Hoening-R. https://arxiv.org/abs/2204.05890. These are instances of the phenomenon I called "redshift", which has now been proved to hold for all $E_\infty$ (= strictly commutative) ring spectra by Burklund-Schlank-Yuan https://arxiv.org/abs/2207.09929.  Moreover, all of $\pi_*(S)$ is detected in $\pi_* K(S)$. So the higher chromatic families in $\pi_*(S)$ are detected in algebraic K-theory of (non-discrete) ring spectra, and should be detected in arithmetic cohomology theories evaluated on these ring spectra.
See Hahn-Raksit-Wilson https://arxiv.org/abs/2206.11208. I am not aware of any extant theory of zeta- or L-functions that reflects these "spectral" cohomology theories.
