Riemann Hypothesis, Primes and Groups

Question

Let $G$ be a finite group $S\subset G$ a generating set $|g|=$ word length with respect to $S$. Set

$$ \sigma(G) = \sum_{H \le G} [G:H]$$

Let $\rho$ be the regular representation and set $A_G := \sum_{g \in G} \frac{1}{1+|g|} \rho(g)$. Then $|A_G| = H_G = \sum_{g \in G} \frac{1}{1+|g|}$, $A_G$ is a normal matrix (and possibly not singular), and $A_G/H_G$ is a doubly stochastic matrix. Let $L(x) = x+\exp(x)\log(x)$ be the Lagarias operator. The "group-theoretic" Lagarias inequality might then be stated as:

$$\sigma(G) \le L(H_G) = L(|A|) (=^? |L(A)|)$$

where $|B| = $ spectral norm. (For $G=C_n$ the cyclic group and $S = \{a\}$ this is equivalent as shown by Lagarias to Riemann Hypothesis.) I have put an $?$ on the equality, since numerics suggest this is true, but I have no proof yet. For a possibly infinte group $G$, Marcus du Sautoy and Fritz Grunewald define:

$$\zeta_G(s) = \sum_{n=1}^\infty \frac{a(G,n)}{n^s} $$ where $a(G,n) = |\{H \le G : [G:H] = n\}|$. For $G = \mathbb{Z}$ this is the Riemann Zeta function. For $G= C_n$ the cyclic group we get: $$\zeta_{C_n}(s) = \sum_{d|n} \frac{1}{d^s} = \sigma_{-s}(n)$$ which was studied by Ramanujan to give (under the RH) an upper bound for $\sigma(n)$.

Now, what does that have to do with primes: Intuitively if
$$\hat{\pi}(n):= \{ p | p \text{ prime }, p \le n \}$$ then $\hat{\pi}($|G|$)$ will determine how much $\sigma(G)$ can grow at most, since by Lagranges theorem for each $H \le G$ we have $[G:H]||G|$, hence $[G:H]$ must be divisible by some primes $p \in \hat{\pi}(|G|)$. For instance by Sylows theorems, we get a tight ($G=C_{p^k}=$ cyclic p-group) lower bound for $\sigma(G)$:

$$\sigma(G) \ge |G| + |G| \sum_{1\le i \le r} \sum_{1 \le n_i \le \alpha_i} \frac{1}{p_i^{n_i}}$$

where $|G| = \prod_{1 \le i \le r} p_i^{\alpha_i}$ is the prime factorization.

So there seems to be a conjectured relationship between the Riemann Hypothesis, primes and finite groups.

My question is, what is the relationship between:

$$\zeta_{\mathbb{Z}}(s), \zeta_G(s) \text{ and } \zeta_{\mathbb{Z}\times G}(s)$$

where $G$ is a finite group and the group $\mathbb{Z}\times G := \mathbb{Z}\times_S G$ is defined in my previous question and is not the direct product. (It turns out, it is the direct product!)

Thanks for your help!

Edit 25.05.2019: All these upper bounds have something in common:

Let $G = C_n$ be the cyclic group.

1) Lagarias inequality: zeta_(Cn) ( -1) = sigma(n) <= L(H_n)

2) Ramanujans upper bound under RH for zeta_(Cn)(s) = sigma_{-s}(n)

3) There is also an upper bound equivalent to RH for the number of divisors:

zeta_(Cn)(0) = tau(n) <= .... some upper bound

4) The "group theoretic" conjectured Lagarias upper bound, can be interpreted as: zeta_G(-1) = sigma(G) <= L(H_G) . 

These are all upper bounds for the zeta functions of some finite group and have directly or indirectly something to do with RH.

Edit 04.06.2019: I found some (unpublished) upper bounds to $\zeta_{C_n}(s) = \sigma_{-s}(n)$ which are equivalent to Lagarias inequality and hence to RH.

Answer 1

Unless I'm mistaken, the group that you're constructing here is an extension of $\mathbb{Z}$ by $G$. It would cause less confusion if you reserved the notation $\mathbb{Z} \times G$ for the direct product. A fairly common notation for this extension is $\mathbb{Z} : G$.

As far as your question on the relationship between zeta functions, a relationship of this type is worked out for any normal subgroup $N$ of a finite group $K$ in a paper of Ken Brown:

Brown, Kenneth S., The coset poset and probabilistic zeta function of a finite group, J. Algebra 225, No. 2, 989-1012 (2000). ZBL0973.20016.

Note that Brown's $P(G,s)$ is (at least in the case where $G = \mathbb{Z}$, haven't completely carefully checked elsewhere) the reciprocal of the $\zeta$ function. Brown also uses a theorem of Gaschütz which is stated only for finite groups. It seems likely to me, however, that you might be able to extend some of what Brown does to your situation.