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.

notthe direct product! Make that more explicit in the post. You have a link to your earlier question about that group but I still think you should emphasize this more clearly. $\endgroup$ – user1728 May 23 '19 at 10:153more comments